I think that aspect you're mentioning there about measurement of what works is so critical and it's hard You know the federal level it may be more monolithic, but you look at states and cities you can say oh wow This city is doing something that's very different from That city and it's working You know Cory Booker was on stage with me at the personal democracy forum last week And you know he was just talking about reducing crime rates in Newark and you know their metric They actually have a metric they're trying to achieve and you know it was very similar in a way to the way You know web developers think about you know some metric they're trying to drive on their website I guess I just want to say even though it's an imperfect analogy event To between laws and programs I think there's a lot of fruitful work that we can do in in working that analogy You know I mean as I just started with a riff on you know Carl's idea that law is the source code of our democracy But this idea that we can in fact start to test and measure the outcomes And I think some of the work that Vivek has been doing is starting to show us how we actually are producing the data that creates The potential for feedback loops about we we did this program and it's working We did this other program and it's not working and the back at least in the it part of the federal government is Starting to close those loops and to kill things that aren't working And I think you know that's part of of where we can take this analogy and Said that the community of people who will study the law the community of people that will implement the law I think it needs to become Sort of it needs to move into the 21st century, you know with with you know with some of these practices of Open access and then as a feedback loops about measuring outcomes just to sorry just to add to the The ability to actually shine light to give you an example I remember when I first came on the job I got handed a PDF document and I was told welcome to your new job and here are 27 billion dollars worth of IT projects that are way behind schedule an over budget out of $76 billion worth of IT projects. I mean that's what I decided. There's no way you could manage By PDFs that are generated on an annual basis. So the analogy I use is very much like a Precision-guided missile where you're getting feedback loops every millisecond in terms of where you are in relation to your target In terms of time and space in the same way with a lot of these IT projects as soon as we started shining light And we made sure that every CIO was telling us every 30 days Where they were in terms of cost schedule and their rating as a result of that We've already seen results as far as at the VA for example We halted 45 IT projects Terminated 12 of the ones that were not performing. We've done a number of tech set sessions where we've seen For example at the SBA. They were spending $1,614 per card. This is your identification card the smart cards were the same exact card You could get a GSA for $240 and What we learned was a lot of people just don't know where they are in relation to the rest of the government or other entities and the ability to really shine light and focus on performance focus on being relentless about Evaluating how we're doing has a huge impact on implementation and The ability to actually think through what is a new legislative process or management processes that we need to implement with Security for example we inert lots of issues and that's why we're engaging with Congress on Rethinking the legislative process So actually I thought this on you know, there we go I actually thought you were going to say that when you took on the job They handed you your head as opposed to the PDF I'm now I'm levitating a little bit because I'm thinking about programs as analogies to The the law and thinking how buggy programs are and how I'm thinking how buggy law must be If the analogy holds up and I suspect it does Do you think that it can be made politically fashionable to have more openness and accountability I think certainly the president has used that Tool in that that sense very successfully so far But it seems to me that success in this domain is going to rely on the public deciding That the political process demands this kind of visibility and that the elected officials can't get elected Unless they subscribe to this notion of open and very visible process So that's an important thing an outcome that I think we need otherwise this idea may not hold up well The other question a very simple technical one. I've noticed that companies are being asked by the SEC To produce their financial information in a particular form called XBRL It's another markup language It's conceivable that if we adopt practices along those lines that more and more information will become machinably visible And it could have two very valuable outcomes. One of them is merely openness and visibility But especially with regard to the risks that people take when they invest in public companies But the other thing is that we may actually be able to use the information in ways We never could use it before because we didn't have the ability to use software to analyze what that's telling us So I'm actually analogy. There is a really good one You know with what Google has done with the Google book scanning project. I know it's been somewhat controversial But there's some fundamental point that was made is that this isn't necessarily about Humans reading all those books. It's about what can you do when computers can read all the books? So, you know, what kind of textual analysis can we do? What kinds of programmatic Comparisons and I think there's a similar potential when all the law is online, you know We actually, you know, you know what we have seen is this explosion of complexity, which makes it impossible I think for the average citizen to understand the laws that govern them and we need We either need radical simplification or lacking that we need some power tools to help us, you know Understand the impact of the laws that we produce and and so I kind of what I'm imagining here is this is the beginning of A new kind of legal science, you know, where we can in fact Study the law using the powerful tools that are at our disposal where we can look and we can discover the conflicts, you know by Scanning and digesting all those laws and saying well this one over here conflicts with that one What are we going to do and we can start to surface those issues we can We can you know build simulations of what are the outcomes, you know There's I think there's a lot of things downstream that would come out of this effort that in fact this also relates to the idea that You know if we make all this free aren't we taking away the economic opportunity of these companies that publish this stuff now? And I think well no actually there's much bigger economic opportunity in creating new kinds of value-added tools for You know studying the law Can I yeah, please yeah, we should really make this interactive with everyone No, no, you guys fascinating absolutely I just wanted to comment on two things one is on the research applications and I think one of the most compelling Cases out there is that legal research today is unable to access the corpus And at a couple of the Laudak of workshops in Colorado and Texas We had some guys from Michigan come down that our doctoral students and lawyers and they demonstrated a class of Applications that have not been possible before like Deciding which legal cases are important based on citation analysis And it was fascinating because they showed us a visualization of Marbury versus Madison over time And it wasn't very important and then about 50 years later It started to get cited and cited and cited and it became one of the landmark cases now with Marbury versus Madison We know that sort of intuitively but with some of the appellate decisions. Maybe we don't There's other classes of Research that can't be done in the law school such as analysis of patent litigation to see whether it varies across districts So that is one of the most compelling use cases I think of making the corpus more broadly available is is to begin analyzing that let me make one more other brief Comment though and that's on this bugs in the legal code issue The vast majority of the law are not Supreme Court decisions. They're not court cases They're not murder cases The vast majority of the law are things like the standard for the amount of lead you can have in paint or What is a toxic chemical or how do you define whether a school bus is driving safely and I think some of those Regulations and technical standards are much more amenable to this this debugging process that That Tim has envisioned so that that's my interjection and I'll let you get back Actually, it's a very it's a very interesting interjection the first Immediate reaction I have is that when you have the corpus in machinable form This still leaves you with the semantic problem of representing what's in that corpus And this is something we're still struggling with if you look at Google Bing and the other search engines They are mostly matching text against text without any real understanding of what any of the words mean Tim Berners-Lee has been struggling with semantic web ideas for about a decade now, and it's still hard I wanted to go back to jump in real quickly before you do that. Yeah, but we also have amazing breakthroughs There's spotty to be sure but where we find that there is some hidden semantic and so for example location great great Example we are now able to surface all kinds of data around Locations and our phones are now able to report our location and therefore we can make all these things relevant And I think in a similar way we don't have to understand everything to be able to understand You know some key elements and so it may be that for example, we're able to identify an outcome You know whether it's you know reduced crime rate Let's say we have laws around incarceration or criminalization of this activity versus that activity we can actually go and say well Let's look at the presence of these laws in these 50 states and then looks let's look at the outcomes and the meaning can come Sometimes work back from the outcome No disagreement with that at all it when you where you have a an organizing principle Like location or time You can apply it in very very general ways to the data you have I'd be a little concerned about correlation versus causation Sorry, you should get back to the original point I but I do want to pick up on something that Vivek mentioned and be very specific about it You mentioned wine is a subject about which I'm quite interested because I have a modest wine cellar Which is intended to be consumed As opposed to admired right so the point I want to make is that I tried to ship some wine to a friend of mine in Chicago Earlier in the year and I made the mistake of going on the web to find out what the rules were For movement of this product from one state to another and I discovered an entire matrix It's a pair wise thing and the states have different rules Paralyze rules so it's an n-squared problem So I was going to suggest to you that if we were looking for a Particular poster child problem space to deal with it would be Wineshipments from state to state and if we figured out a way to simplify that we would have done a really great thing It would be a work thing a worked example of simplification So people would be able to share this wonderful product I think we ought to hear from people sitting around the table who've been thinking about this stuff Mike We have microphones. This is the controlled substance so to speak This gentleman had his hand up right there. Yes Well, just to carry your theme I'm paul ver kyle from the administrative conference And I'm gonna be summing up a little in a little while Which is gonna be an impossible task, but I think these are brilliant presentations and for someone who is a new government Official and trying to restart an agency who's concerned about this. It's it's quite gratifying to hear from you With respect to law and how you can determine whether a law is effective or whether you can debug it You do have to use your wine example the 21st amendment. We decided long, you know 1918-19 that we were going to debug the problem by eliminating it Well, that was not very successful and it has taken us and then we changed our mind, you know after we after the the ineffectiveness of the law became so obvious Where we created a whole criminal enterprise? We decided we would went on do it and then we're back and it only is last year or two years ago in the court 5-4 they decided you could send your wine To another state and the state laws couldn't prevent you from doing so so I don't think it's so complicated Wow, that's just me but it's another problem with law just to make the point the law isn't only about what's on the books, right? It's about enforcement and it's about a discretion in enforcement and so that is why it's so hard to analyze It's not about getting everything on the you know It'd be nice if we could have all every law and we could look at them and find out flaws But that only begins opens the discussion the real discussion the discussion we have to have in policy making You know in government is how do you enforce the laws and who does? There's a beautiful example of a case in the Civil Rights Commission where there is in Louisiana a justice of the peace who refuses to marry mixed-race couples and He was told that in 1967 the Supreme Court decided that was Impermissible and he's still doing it this gentleman needs a microphone John stolen the mic. So there he is It's like I'm round. I'll pretend to be Ralph Reagan for the moment So I have a couple of thoughts from really great presentation One is that I think some of the complexity in the law Is a reflection of the fact that in governmental systems There are values being expressed that are not just about efficiency then raise the SEC as a as a as an example of having the corpus in machine readable form in terms of of the of the Filings that are being taken for public companies, but the SEC now right now is struggling with flash trading Which is a different sort of problem of corpus in machine readable form, which used to be I think in in in sort of in in the pre ability to really analyze Data in in in virtual micro seconds used to be thought of but perhaps as trading in front of your clients now Enterprises are creating vass numbers of trades and and wealth to their to their own companies To their own trading systems by do by executing flash trading programs But I think to some extent there's a systemic risk that comes from that and to a different extent there's a sort of a Latent unfairness to to average Traders in a system that permits the ability to to Capture those rents by being analyzed that that analyze that corpus so in government. I think a lot of times the complexity is added by by a set of values that aren't just aimed at New product or efficiency, and I think we need to kind of introduce that into into the conversation and The other Observation I think I would make is that raised the issue of sort of doing what's what works in the internal government systems To try to enhance the productivity of the enterprise of government itself, but ultimately I think the real test Is for the clients receipt or the delivery of government service, which I think is even a more challenging Problem or more challenging question it lends itself. I think to what Tim was talking about If you imagine the education reform movement and the use of data, etc. It lends itself to trying to debug What's working and what's not working, but it's a it's a harder problem I think when you're when you're thinking about the Delivery of government service at the consumer side than in terms of in using better business practice Practices on the inside of government to enhance and make more efficient The way the government is actually working on a day-to-day basis And and I think Vivek you and and the crew over there are really I think operating at both Levels, and I wonder whether you share the thought that one's easier or more difficult or what your experience is on Actually trying to see whether The delivery on the consumer level side is being enhanced So so I think If we look at healthcare for example one of the things we realized what was this principle of Shining light on the performance was What's game-changing in terms of allowing policymakers managers to make different decisions? What was interesting is as we're looking at the healthcare system one of the challenges we found was that there was more Transparency around actually buying a camera online and the ability to compare aperture size or cars online and the ability to compare Speed zero to sixty then there was one hospital to another one doctor to the other And so we're looking at a body of work on how how can we rethink? CMS Literally the concept of dashboard applied to that model or what happened with a community health data initiative That we're working on at HHS where literally we just put the data out there and third parties Started taking that data and finding intersection And I'm a true believer that intersection lies at the value of whether it's multiple Disciplines or data set so they literally took the geodata Combined it with funding data and layered on top of it outcomes data and suddenly this map emerged around poverty levels in Texas and Which county how much money we were spending? What were the policy decisions that were being made? I agree with you. That's it's a harder issue to solve But I think the only way we're going to be able to solve it is by unlocking a lot of this public data And I also think there's some really interesting Advances that have been made geolocation being one of those because it's very important It grounds you the other advance that needs to be made is around this common architecture on funding where are we spending money as a society and Laws play obviously the role in terms of how money is allocated to solve some of the toughest problems We face from public safety to education to health care But but it is a more difficult problem because we don't necessarily always own all the information that we need And we don't have what I call Transparency at the atomic level which is at the point of delivery. How do we get that? Atomic data to be able to then make general Observations whether a program is working or not and where we should be doubling down versus where should we be divesting? Could I know you have a hand up so we need to get the microphone down there? I wanted to make one small observation a lot of the phrase transparency. I think is often interpreted as How to expose those buggers who make bad decisions or do nefarious things? There's another utility and transparency that showed up in the internet related to something that Tim said And I think Vivek also and that's where when you share information You allow others to help you evaluate analyze it and create tools for using it So there is a very positive potential That arises out of making information available and that is that third parties can't help Evaluate it better. So there's a collaborative element in all of this Which has been fundamental to the internet's evolution the millions of people who do things? To make information available or make it more easily discoverable or more easily used is what has allowed the internet to become What it what it is today from its very humble origins? You have a question? Yeah, hi Rob Schecter As far on the issue of now that we have these Statutes and codes machine readable, you know, what can really happen besides just presenting it nicely Something that I've been really interested in implementing and working with is the Library of Congress subject headings which people are beginning to discover and think about new purposes for and This is just a great it's online in a couple different machine forms XML and other And so basically most people see it as a really interesting controlled vocabulary How do you describe law? What are the various subtopics and subcategories and for me? I find it really interesting because every state Oregon we break up the laws and we talk about them in different ways and It's every state has done that in New York. They have something called mental hygiene law I don't maybe a New York lawyer can explain that one to me Yeah, is that right we don't have that in Oregon so Yeah, so so what I imagine is and this wouldn't even be very hard like an afternoon project per state like a per like software engineering terms a per state adapter Translate mental hygiene in New York into the Library of Congress term and then if that existed for each state all of a sudden You have cross-state Searching and knowledge really and there and people have already made I believe this kind of adapter for France and for Canada Canadian law as well into Library of Congress subject headings our Library of Congress for these other countries or I believe these other countries have actually adopted them Which is interesting And I had one other as I had a concrete Example of debugging the law which yeah can be hard and there's there's many subtle problems And I think this will become more common is our sorts of tools become more common on the web one issue in law is Civil penalties that sometimes accrue to someone not just criminal penalties when they've done something bad if you've committed been Can if you've been committed for you've found guilty of robbery? maybe your landlord is allowed to kick you out now, but that's like a civil sort of a law and Sometimes due process concerns can come up because the burden of proof is different well Once you have a collection of statutes and laws and each one knows into what sort of category it is you can begin doing searches and comparing civil section of penalties versus criminal penalties And or white collar versus other kinds of crimes And I found that the this data is already there in the sense that the legislature for a hundred years in Oregon at least they've created this wonderful index with terms like if you want to search for dog You'll say ck9 and to me this is a tag cloud They've been working on a tag cloud for a hundred years, but they didn't consider it like that It's this awesome ontology and so this lets nice intelligent queries be done You know how many links in and out are there to the to civil penalties versus criminal, etc. You know this whole Direction you're talking about makes me think a little bit of what we see in the private sector with things like Tax you know where turbo tax Let's you know an ordinary person File their taxes more efficiently because somebody has done the job of understanding The laws and regulations and turning it into an application And obviously tax professionals still you know work the more complex cases But you've actually automated some of the simpler cases and I do wonder if there's some real interesting applications You know once we start to be able to look at all these laws, you know, can you actually build? better Interfaces for citizens to actually do things that right now are very burdensome, you know getting permits You know, why do they take you know two months six months, you know Because there's a long complicated manual process and it may well be that there is an automated process where you go Okay, we have understood all the the things that we have to do here And can we now build a better interface to all the different places? I know there's one group for example that's working on this in the In the social sector, you know, you're eligible for 40 different poverty reduction programs And you have to apply 40 different times nobody does and they're trying to you know The project called intake one where you know, you basically fill out one form and it figures out how to navigate the You know the the various agencies to get you the benefits that you're entitled to and I think it's sort of a whole set of applications I think that start to fall out of thinking about this as a System that you can start to engage with Via computing as well as as via, you know people sitting down and just reading it There was a series of presentations on access to justice particularly at the Chicago Kent Law Dock of workshop and that's one of the things that they were most interested in you know The vast majority of Legal proceedings are not done by lawyers. They're done by individuals trying to get into the system I believe in in Florida more than 50% of divorces are not involving lawyers or their pro se and the access to justice people are Very interested in making the law available and then simplifying the process of letting people deal with the legal system And that's one reason and I think people like professor tribe We're here this morning and a lot of the access to justice people are very Interested I hate to cut this off because again we could go all day, but I think we're ready for our final panel at this point I want to thank you all very much. This is really fascinating. Thank you very much Out of the lawyers in Florida would be involved in divorce cases of their own


Okay, thank you very much for the invitation. Um, yeah, so I'd like to talk about Vanda that numbers. Only I can't advance my slides there we are. So, there's a very famous theorem of Vanda that from the 1920s, which in its two color version states the following. So take integers R and S. The number, which I'm going to call WR comma S in this talk, which has the following property. If I take an integer N that's bigger than that number, WRS, and if I two color, the integers one up to N. So my colors are always going to be blue and red, then either there will be an R term arithmetic progression that's blue, or an S term arithmetic progression that's red. So that's Vanda van theorem from the 1920s. Now it will take WRS to be the smallest number with this property. Obviously, any bigger number would also work. So define WRS to be the smallest number with this property. That's called a Vanda van number. So for the most part, trying to estimate what these are is very much an unsolved problem. And a particularly notorious case is when R is equal to S. So you're looking for either a blue R term progression or a red R term progression. And there the bounds are out by several towers of exponentials. And in fact, until Tim Gowers did his famous work 20 years ago, the best known upper bounds for those numbers were enormous sort of Ackerman type function or certainly very large towers of twos. So these are sort of notorious open questions. I'm going to be interested in what's called an off diagonal Vanda van number, which is where one of the numbers is three so you're looking at blue three term progressions versus red K term progressions. So W3 of K is the smallest number, such that if you color one up to N, if N is bigger than that number, if you color one up to N blue and red, there's either going to be a blue three term progression, or a red K term progression. So what's known about this, the best known upper bound is not too bad. It's a little bit better than E to the K. And that was done very recently by by Thomas Schoen. And that also would follow from the celebrated recent work of Blumen seesask on roster on three term progressions. So that's the best known upper bound at the moment. Here C is some very, very small positive constant, a little bit better than E to the K. The best known low bounds in the literature are much worse than this, basically K squared, in fact slightly worse than K squared K over log K squared by Lee and shoe. And there's quite a lot of numerical calculation of these w three K numbers. In fact, it's a favored sort of environment for people who do things with sat solvers to test what they're doing. So actually, there's numerical evidence, numerical values for w three K up to K equals 20, and then lower bounds which are speculated to be the right numbers for bigger care as well up to about K equals 40. So I'll show you those on the next slide. And a lot of names associated with that endeavor that I've written down there. So let's have a look at this data. This is a table of known values of w three K. And once you get up to K equals 20. These are in italics, which means that they're rigorous lower bands. And I think they definitely conjectured to be optimal. So that is to say actually the correct values up to K equals 30, and maybe a little bit more speculative after that. So these values should be taken with a pinch of salt. And I believe that it's certainly very reasonable to conjecture given this data. In fact, almost the only thing you could conjecture is that w three of K grows like K squared. So I first heard this question from Ron Graham 15 years or so ago. And I think Ron said to me, I think w three of K is roughly K squared, certainly bounded above by constant times K squared.  I immediately said that cannot be right. And here's why it can't be right. All we need to do is to take a blue set B that's got no three time progressions. So that's already half the battle color everything else red. And then with luck, that red set won't have any long progressions. Now, if that red set looked like a random set, whatever that means, then we'd expect the longest progression in R to have size about N over the size of B up to logarithmic factors. If instead of progression you you think interval that's a sort of very well known principle if you've got a set B inside one up to N, the gaps are going to be on average of size N over B. So basically there won't be any too much longer than that. And there are many examples, many famous examples of these blue sets B with no three time progressions of size much bigger than square root N. And therefore, if this heuristic that I've put up there had any merit, those red sets would have no progressions longer than about square root N, which would disprove Ron Graham's conjecture. All you need to do is try it on your favorite examples of large blue sets with no three time progressions and here are some the folklore example that consists of numbers with just zeros and ones in ternary, which also came up as an IMO problem actually in 1983. So that's got size bigger than route and it's got size n to the log two over log three, and it's an easy exercise to check that that has no three time progressions. It's much better than that very famous example of their end, based on a slightly earlier example of Salam and Spencer gets better than any power of n less than one so n e to the minus constant root log n. This comes from taking the lattice points on a sphere so you can observe that if you take the points on a sphere that has no three time progression in fact doesn't even have three points on the line. So if you get that down in a suitable way down to the one dimensional integers, then you get a large set for your three time progressions. And there's a variant of that construction due to Julia Wolf and myself in 2007. So there's three things that you can try here that will give you very large blue set. What you need to do then is just to show that the red set the complement of that blue set has no long progressions. So that was the thinking behind my saying to Ron Graham that I thought his question was ridiculous. Unfortunately, I have a question in the audience. So, recently, maybe you want to unmute and just ask away. I just read the question. So there's a connection PD wonder that numbers and Ramsey numbers. That's the question. Well, the definitions are somewhat similar. Ramsey numbers, the most natural Ramsey number that's of a similar flavor to this would be our three K where you're two coloring the complete graph, and you want either a blue triangle or a clique of size K that's red. But other than that, I don't know of any direct connection. Somehow clique is a somewhat quite a considerably less structured object than an arithmetic progression, I would say. There's similar definitions but no direct connection. Yeah, so back to this sort of simple minded attempt to refute Ron, Ron Graham's conjecture. So the first thing to try is the folklore example of the set with zeros and ones in its ternary expansion. And then you immediately see that there might be some difficulties here, because the complement of that said contains enormously long arithmetic progressions in fact it contains everything that's to mod three. That means progressions of linear size. So this is going to be a terrible example for the van der Verden number problem. And unfortunately the Salem Spencer and Baron examples also don't work for somewhat similar reasons. I won't tell you precisely what those examples are but they're all consists of things whose digits are somehow constrained in somewhat small bases. So both have digits restricted to just zero up to D minus one in some base to D minus one. And here D is going to be quite small. So it's something like order log n. So again, such a red, the complement of that set B, which is going to be the red points has very, very long arithmetic progression way bigger than the square return. Okay. So here's a couple of Julia Wolf and me. I want to talk about this in a bit more detail because actually it's the starting point for the construction. I'm going to talk about later. So here we take some integer dimension D and look at the tourist T to the D. And let pi be the natural projection from R to the D to T to the D. So that's actually notation I'm going to keep throughout the talk. And D, consider a Euclidean annulus of radius one quarter, and of this rather small width, enter the minus four over D. That turns out to be a convenient choice. I'll explain a bit more why in a minute project it down to the Torah. So you've got this little annulus sitting inside a tourist in D dimensions. And now we're going to create a coloring from that. So what you do is you pick a random theta vector theta in the tourists and look at the orbit theta two theta three theta and so on. And let the blue points be the set of times N at which that orbit hits the annulus. Okay, so I'm going to put a little picture up. Let's see the complement of that set of points. So here is a picture. So that is supposed to be a tourist. I've drawn a two dimensional one. And the annulus is there quite clearly. And then you're going to take an orbit. I think the one I drew starts from here. Red red red red red. And then when it hits the annulus it's blue red blue and so on and then it sort of wraps around up to here comes down like that. And then you go over here. And then probably down like this. So this this particular segment of an orbit only has two blue points on it. Everything else is red. So that's the construction. And why you'd expect that the number of blue points here, because these are as picked randomly, it's going to be proportional to the volume of that annulus times N. So that's roughly end to the one minus four over D. So as soon as D is at all large, say D is bigger than eight. The number of blue points is going to be bigger than square root N. And again, you can hope that this red set shouldn't contain any long progressions. But before I get to that, I've got to explain to you why I hope that this set B doesn't actually contain any blue three term progressions. Why am I taking this annulus? Well, this is just a sort of slightly padded out version of the fact that a sphere doesn't have three points in a line, just a sort of thickened up version of that statement. So A is this annulus. And then we have the parallelogram law, simple fact from Euclidean geometry. And so if I've got three points in that annulus A, and I'm now thinking in R to the D. And if they're in arithmetic progression, then the common difference of that arithmetic progression has got to be absolutely tiny by this parallelogram law. It's got to be its L2 norm is at most N to the minus two over D. That's in R to the D. But when you push things down to T to the D, because the radius of this tourist is one quarter, you're not going to gain any extra progressions by sort of wraparound effects. So here's a picture of this. The only way you're going to get a blue progression is if it just sort of grazes against the very edge of this tourist like this. And if the step, the common difference is very, very small. So, although inside this annulus because it's an open set, but definitely are collinear points. There are, they've got to be very, very highly bunched together. If there was a three term arithmetic progression and N plus DN plus 2D of blue points, then theta times D, where theta is this randomly chosen rotation on the tourists would have to have very, very small norm where this notation here means that all of the coordinates, Modulo one, that's small. So all of the coordinates are at most N to the minus two over D. And if you fix D, and remember that thesis chosen randomly, the probability of this happening is absolutely minute. It's actually enter the minus two. So we've got a theta times D sits inside some box of width, enter the minus two over D the volume of that box is enter the minus two. And then you can just take a union bound over all of the possible choices of D of which there are most N. And you see that the chance that there's any blue three term progression at all is absolutely minute. So this is why annual I are a good thing to choose. They're basically just slightly thickened up spheres, spheres have no three term progressions. So I feel I have very, very few three term progression. So that's why it's a sensible construction to consider. So what about red progressions. So the red points are the complement of the blue points. Let's just recap some of the notation quickly. So I've got this random rotation theta on the tourists. Let me write theta one up to theta D for its coordinates. And then an actual small piece of number theory, I can apply Dirichlet's theorem. The most simple thing that one learns about time approximation. So there's a non zero D less than square root n, such that the fractional part of theta one times D is at most one over squares. Now, theta is a randomly chosen vector. You certainly so theta one is also random. And you expect the sequence theta one to three to one three to one and so on to equal distribute very, very rapidly module one. And so certainly before too long you'll find an n naught such that these are one times n naught is close to a half. But then if I take the progression that starts at n naught and has common differences D that I found using Dirichlet's theorem. Well that's going to stay near a half for a long time. So one times n naught plus LD, here L's an integer will be a half plus something that's on the order of L over root m. So this is a picture of what happens. So this progression starts over here, it's x coordinate if you like it one coordinate is kind of close to a half. And then it just stays there for a very long time sort of wraps around and around and around, but it never hits this annulus. So it's it's first coordinate just is stuck near a half. And that happens all the way up to L being close to squaries event. So unfortunately, this construction by Julia Wolf and myself doesn't work either. So you can use this construction to get a large set that's free of three 10 progressions color it blue. But by Dirichlet's theorem. You're still going to be left with a red progression whose length is is basically squaries event. So at this point, I have to admit that I don't know how to disprove Ron Graham's conjecture it's possible that w three of K is bounded by constant K square. And in fact, I went ahead and conjecture this myself will be reiterated his conjecture in print. So everything so far is consistent with the fact that this off diagonal van der bedden number, the smallest n such that if you two color one up to end, you're guaranteed to have either blue three 10 progression or red K 10 progression. That could be quadratic in K. However, it turns out that that's not the case so this is the new theorem that I want to talk about today. So actually w three of K does grow quicker than any polynomial in K. And in fact it grows like basically K to the log K to the one third. So in particular it grows faster than any polynomial in K. So, let me talk a little bit about how that construction works. And I showed you three possible constructions there was the one based on the digits in ternary. And then there was one based on the baron construction also really really didn't work. And then there was the one based on my work with Julia Wolf which did a little bit better in that the red progressions at least weren't linear in size they looked as if they were down to square root and inside that didn't prove that there were no longer progressions, but the ones that I found only had size square 10. So what was the problem with that construction that prevented you rolling out progressions of length through 10. The problem was that I took this annulus of radius of quarter pushed it down to the tourists. The problem was that the complement of that annulus contained cosets of co dimension one sub to right. In fact, in that case it just just the sub tourist T one equals a half of coset T one equals a half. And then you could find orbits theta P where P is a progression of length square root n using Dirichlet's theorem you can find an orbit of that length, which concentrates really near such a sub tourist. And so that progression is guaranteed to be read. So the first new idea is that is that taking one annulus perhaps we can take a union of several annual I am slightly smaller radius which are called row. And perhaps if you choose those premises judiciously, you can block sub to right so if you take that union of annulus annual I. There's going to be no co dimension one sub tourist in the complement. And hopefully there'll also still be no blue three term progressions. So that's the, that's the basic idea is take more than one annulus instead of just one. So let's take a picture of this. These are only lie of radius row where row is something like one 12 in this picture. And the way I've drawn them. There's no obvious way of sort of threading a co dimension one sub tourist through them without somehow help however you try and do that. You're going to run into one of these annual life. So there's no obvious way through. And then the construction once again is going to be, I take an orbit theta two theta three theta. I look at where it hits this union of annual I, and when it hits them, I color blue. And when it doesn't hit them, I color red. Okay. So, I first of all need to justify that I can that there's a chance of doing this in such a way that I block. Co dimension one subject. So in other words, there are enough of these annual I that I can't thread a sub tourist through them without hitting them. But they're not so many of them that there's guaranteed to be a three 10 progression for trivial reasons. So you've got to make sure that you can balance the parameters to achieve these two somewhat contradictory aims. And I'll just tell you, it's probably the most mathematical part of the talk. We're going to in too much detail. Try and explain to you why that can be done. So here's how one should choose the parameters you fix, fix a dimension D, which is going to be the dimension of this tourist. The radius of those annual I will be D to the minus four there's quite a lot of flexibility there but it should basically be a power of D one over a moderate power of D. So we're going to look at. First of all just one annulus a of radius this row in Euclidean space. And now push it down to the tourists. And now I'm going to pick a number of random translates at that inside the tourists. And the right number of random translates to take a game there's quite a lot of flexibility but we'll take D to the D of them. And then we're going to consider D to the D of these little annual I of radius row. So all the annual I congruent, and we translate them around D to the D times, and we'll call that set S. And then the construction which I drew on the previous slide, pick a random theta in T to the D, and to find the blue points to be the set of return times of this orbit theta two theta three theta and so on. And then we'll call that set S, which is a union of annual I, and the red points to be the complement of the blue points. So that's the construction. Okay. So why can I hope that there are no blue three term progressions. Well this is essentially a fact from to do with coding theory intuitions really. You can pack a lot in high dimensions you can pack a lot of these points in, while still keeping them highly separated. And actually I need them to be a bit better than highly separated I need the sort of difference of them to be highly separated like this. But I claim that with high probability. So the three term progression of these D to the D to the D centers of these annual I are not only not in three 10 progression but they're very far from being in three 10 progression so they're separated by 10 row. And that means that if you put a little annulus around each one. Well the only way that you could end up with a three 10 progression is if it's completely inside just one of those annual you can't have a three 10 progression that's across different. Okay, so that's the first claim. So the progression in the set S has got to lie entirely in just one of those annually, it can't sit across different time. And therefore, by the parallelogram law as I talked about before, it's got to have very very small common difference. So that will eventually lead to that being with high probability, no blue three term progressions. So as I said this is reminiscent of simple sort of packing bounds that lead to the simplest bounds in sphere packing really. I want I two and I three. If they're not all equal, then this XI one minus two XI two plus XI three is uniformly distributed on the tourists. Therefore the chance that it's sub norm is at most 10 row is very small. It's at most, it's 20 row to the D. You can work through the parameters carefully and check that the probability of this happening for any one of the m cubed choices of the indices I one I two and I three is minuscule. So that's why there are no blue three 10 progressions. What about the red progressions. So I need to make sure that this union of annual I blocks subtory so I need to make sure that there isn't a co-dimension one sub torrents in the complement of those. So for the blue progressions. I couldn't afford for there to be too many centers XI. And this is pulling in the opposite direction. So here, I need there to be lots of the centers XI to make sure that there's a chance of blocking subtory. So it doesn't even mean mathematically to block subtory. But what it really means is that if you take a character from additive character from T to the D to our mod Z, then it's subjective on that set S. So in particular, Kai could be the character that just takes the first coordinate. So we're saying that there's no fixed. If you look at a sub to co-dimension one subtory is with just a fixed value of the first coordinate. Intersex S. So S has blocked that particular co-dimension one sub torrents. Very rough sketch of the proof. Well there's only a bounded number of these size I need to worry about because as soon as say XI one is large, it will sort of stretch out the annulus to the point where it automatically covers everything. Just one of these annual I pie a it has radius row. If this character has got a large coefficient, it will sort of stretch out the annulus and its projection mod one will be everything. So there's only a sort of bounded number of them I need to worry about. And then I just need to check that the centers X1 up to XM under such a character are dense mod one. And then I will, the rest of the annulus will push forward to cover everything. And this is what's called a bin packing problem. So essentially the problem is for each fixed psi, these psi applied to X1 up to psi applied to XM.  The number of bins is one over one over the size of each bin is row. The basic point is that I just got way more balls than I have. And so the probability that I've got at least one ball in every bin is overwhelmingly high. So, and then I can just sum over all of the possibilities for XI which is row to the minus D. And again do a calculation and this calculation wins by absolutely miles because I basically got an E to the minus D to the D here versus just a D to the D.  So it turns out these two contradictory aims. So choosing few enough annual I that you don't create three time progressions, but enough annual I that you block all co-dimension one sub toray. They can be achieved simultaneously by the right choice of parameters. Now, I've been talking about this co-dimension one sub toray. They came about because I applied Dirichlet's theorem. Now you can also apply Dirichlet's theorem with more than one variable. And then you have to start worrying about co-dimension two and three and four sub toray and so on. But actually with the right choice of parameters, you can block a whole load of those all at once. So the construction I just described it gives you a blue red coloring of one up to one with no three time progression. And there's no obvious way of making a red progression of length square root n. So where obvious I mean you cannot use Dirichlet's theorem to make a red progression of length square root n. Because I blocked all of the co-dimension one sub toy. And as I said just now, by making sure that higher co-dimension sub toy are blocked. You can also prevent more higher number of variables application of Dirichlet's theorem from creating progressions of length n to the one third n to the one quarter and so on. So this gives you a construction that at least has a chance of having no blue three time progressions and no long red three time progressions. But of course it's nothing like a proof that there are no such red progressions. All I've done is I made a construction which defeats somehow the most obvious way of producing long red progressions that might very well be other long red progressions. Okay, so let's talk a bit about how you can hope to show that in fact there are no such things. So it's very natural to fix a value of r and focus on rolling out red progressions of length n to the one over r. So maybe I just as an aside. So I said that to block progressions of length root n you need to look at co-dimension one sub toy. But then you can apply Dirichlet's theorem with two variables, potentially to find red progressions of length n to the one third. And there's no, there's nothing in between these two things you either apply Dirichlet's theorem with one number, or with two, but nothing in between. And this I think is probably behind that data that I showed you right at the start of the talk, where you see quadratic behavior. So what I sort of expect is going to happen is that you see quadratic behavior up to a point, and then it will jump to something more like cubic behavior as sort of small numbers phenomena go away and it starts being more efficient to look at constructions like this, where you're blocking Torah. And then there might be another phase change later on. Anyway, for this reason it's very natural to introduce a parameter r and try and study progressions of length n to the one over r. So anything about what D is. So I'm going to perform that union of annual construction that I showed you with a specific value of D that's tailored to the R that I'm interested in. So you can think of our being three if you like. So D will be some constant times r squared. And to perform the annual construction with a randomly chosen theta. So Dp a progression of length x inside one up to n, and I want to show that it's not all red doesn't all get colored red. So in other words with high probability it contains a blue point. So how are we going to do that. Well I'll do it in two stages. First of all, use the first half of that progression. Not to hit an annulus that's too much to do all in one go, but to get inside one of those annual. So we'll first show that if you follow the orbit theta P. Then, by the time you followed half of that orbit, you've gone inside a little ball of radius row over 10 remember rose is the radius of the annual I. So you're, you're right inside one of these annual I, and just for simplicity of notation let me suppose that the center of this annulus is the origin zero. So you're in the middle of one of these annual I and then the natural thing to do is to show that you sort of have to hit the annulus on your way out as it were. So we're going to use the remaining portion of this progression P to actually hit this annulus. So we're going to use the X, the projected version of the set of points X, whose our two norm is between row minus 10 to the minus four of D and row. Okay. So these are two quite different flavors of problem the first one is what I call a core scale equal distribution results, you've got an orbit, a linear orbit on the tourists, and you want to show it hits some ball. So you've got the numbers of reasonably large size, and you can apply the kind of techniques you'd expect to apply in this sort of equal distribution problem so for your base techniques exponential some techniques to show that under suitable conditions. You do get inside one of these balls. And then the second stages is a much finer scale result. The annulus is very, very thin. You cannot just go and apply the core scale for your techniques to try and locate a point inside one of these. So you've got to do something different. And I'm going to talk mostly about stage two in the rest of the talk. So what kind of a problem is this. So let's suppose I'm at zero. So that's what stage one has given me. And now I've got this orbit. And this is actually that the common difference of P is one so I'm really just looking at the progression 1234 song. And I want to show that one of the points and these are one dots up to empty to D lies inside this annulus. And so the rough kind of flavor of this problem is showing that a certain quadratic form. So the variation here means the absolute value of the fractional parts it's the norm on the circle lies very close to row squared. So the basic flavor is I've got something like a quadratic form and I want to show that it takes a value in a very short interval. But of course the, the obvious problem is this isn't a quadratic form. It's a kind of bracket quadratic form. So that's going to obviously make things a lot nastier to deal with. So the problem is to show a certain quadratic form where quadratic form means bracket quadratic form takes at least one value in a very small interval. So the first step is to take this nasty bracket quadratic form and turn it into something that's really an actual quadratic form. And now we use some more sophisticated geometry of numbers that beyond Dirichlet's theorem. So basically Minkowski's second theorem. This is a fairly well-trained path in, in attitude combinatorics and geometry of numbers. The set of n for which all of the fractional parts of these n theaters are small, let's say at most 110, has a multi-dimensional structure. So this is a well-known fact from geometry of numbers. So more precisely it means that if you look inside the set of such n, it contains a very large multi-dimensional progression. It's a kind of grid based on points n1 up to md plus one with side lengths capital Li. And the product of those side lengths capital L1 times up to capital LD plus one is comparable to x, which is the total number of points and consideration. So I really most a large number of the points and for which this is true, which all their fractional parts at most attempt are captured by this multi-dimensional structure. So this goes into misprint. That's really what it is. This is somehow an artifact of the fact that we're working over the integers in additive combinatorics. This sort of thing is usually done in finite cyclic groups modular prime. And there is D instead of D plus one. Okay, so you can put a multi-dimensional structure on things. So here's roughly what it's going to look like you can imagine this progression P. It's going to look multi-dimensional. By passing to this new basis that geometry of numbers gives me. Here it's two dimensional. That's wrong. It should be three dimensional, but a three dimensional drawing looks a total mess. And when it hits this annulus, I've colored things blue. So that's the sort of situation I have. So what's the task in this new basis, which geometry of numbers has given me. I have a genuine quadratic form, no brackets anymore in D plus one variables. And I want to show that this quadratic form hits a user specified, so a sort of given short interval. Now representative case just to get one's head around the sort of nature of this problem is that all of the lengths L one up to LD plus one of this generalized progression multi-dimensional progression. They're all the same. So in the generic case, you have to worry about widely differing lengths as well, but just it's a reasonable representative case. So those lengths would all be about X to the one over D plus one. The coefficients of that quadratic turn out to be fairly small. So in particular the values of that quadratic are bounded in size. And then the length of the interval that I'm aiming for which is enter the minus four over D. So the parameters. There's probably too many to remember here but I chose D and X was like enter the one over R. The bottom line is that the length of the interval that I'm trying to hit with this quadratic form is about L to the minus four R. So remember here R was. I'm trying to rule out read progressions of length and to the one over R. And I said if you like you can think about our being three. Of course this is L to the minus 12. The basic problem is I've got a fixed quadratic form in some large number of variables. I want to show that it hits some interval of length L to the minus 12. As the variables in that quadratic form range up to L. So you can forget the rest of the construction and all of this picture now let me just restate what the problem that I've reduced things to is. So R is an integer. I said you can think about RS three. D is some large constant times R squared. Take a quadratic form on Z to the D plus one with coefficients of size one upon L squared on the box out of the power D plus one. This Q is bounded in size. And the problem is just to show that it takes at least one value on any given interval of length L to the minus four R. Okay, so that's the nature of the problem. It should be true by miles because the number of different values of Q is L to the D. But D is big constant times R squared. And so you've got all of these many, many values of Q. It's bounded in size so by the pigeonhole principle that the sort of average gap between values is is more like L to the minus D. And D is massively bigger than four R. So it should be true heuristically by miles. So that's the naive heuristic can you take so many values that it really you feel like it ought to be the case that it hits every interval of length L to the minus four R with masses of room to spare. It's a bit too naive. And the reason is that there's an integrality problem. So Q could, if we were unlucky, be basically rational. So it could have all of its coefficients integers over some fixed Q. So Q here, maybe just a little bit bigger than L squared, just to make sure that this is possible. If this happened then the values of Q would obviously be separated by one over Q the values of little Q the quadratic form is separated by one over big Q. And if this big Q is moderate in size, maybe L cubed or something. Well then the values of this quadratic form are going to be very separated and there's not really any reason to imagine that you'll hit any particular interval of length out to the minus four R. So as I've stated, it has a negative answer, because potentially, I mean, Q could be zero, but there are, there are other worst things that could happen, it could be close to a rational form essentially. So how are we going to get out of this? Well the idea is to use randomness. Q is not some fixed just fixed form that's out there, it depends on the choice of this random rotation theta. So there's a chance that I can show because theta is random. There's maybe a chance that I can show that almost surely Q is not integral in any vague way. Unfortunately, that that seems to be just impossible. I have no idea how to really control the behavior of Q as a function of beta. So the second key idea of the paper is to introduce an enormous amount of randomness elsewhere in the problem. So you can replace these spherical annuli by ellipsoidal annuli where the eccentricity of the ellipse is chosen randomly. And you indeed mentions so the eccentricity of an ellipse is a kind of the configuration spaces d squared dimensional essentially. So you've got this enormous parameter space of random ellipsoids. Instead of taking the identity matrix you perturb it by a little random E. And you define your annulus to be this. So it's an ellipsoidal annulus. All of the arguments that I mentioned earlier to get to do with the parallelogram law and so on they all apply just almost exactly the same with these ellipsoidal annuli. So nothing's been lost there. But you've now introduced an enormous amount of randomness into the problem. And this quadratic form Q that I want to show has small gaps depends on this random E. And if you work hard you can actually show that as he ranges overall choices of eccentricity of this ellipse. Roughly speaking, those quadratic forms range uniformly over all quadratic forms with bounded coefficients. So there's a lot of technical detail that I've suppressed there. But that's the basic point. So I certainly don't know how to show anything like that just using the randomness of theta and spherical annuli. But if you introduce the random ellipsoidal annuli, you get a far richer space of quadratic forms. So by doing this I've reduced the problem I had before about showing that a specific quadratic form. It hits every interval to the same problem but for a random quadratic form essentially. So here's the theorem that we need to prove. B is just some exponent. It's related to that R that I mentioned before but it's not too important. And I claim the following proposition. So suppose that S is bigger than B squared. And let me take lengths L1 up to LS all of size roughly L. And we'll do some quadratic form by choosing the tuple of its coefficients Aij uniformly at random. And we'll do it so that those coefficients are sampled uniformly from minus 100 up to 100. With the exception of the diagonal terms which I want to be strictly positive. So they're going to be sampled just from 10 up to 100. So write down that random quadratic form. Then the claim is that it comes very, very near to any fixed value that you that you want. So here is the write down the event that the values of this quadratic form on the grid with side lengths L1 up to LS. And let Sigma be the event that that discrete set of points comes within L to the minus B of every point between say 0.9 and 1.1. So you want to you want to get a value of this quadratic form that's very near one say so you can actually get it dense in this interval. So that won't happen all the time because Q could be it could be rational. The theorem is that it happens outside of a very small exceptional set. So, in fact, the probability that this doesn't happen is a really small power of a really big power of one over L so L to the minus BS essentially. Now this really is quite a small exceptional probability and that's actually important. So this is the main proposition. Okay, so what goes into the proof of that. Well many number theorists when faced with a problem like this I think would turn straight to the hardy little wood method which is indeed what I did. So you, you've got a quadratic form and you want to try and show that it. It attains values on an interval, the hardy little wood method or more particularly the Davenport harbour invariance of this is really well set up to do that. So it's technically a little bit demanding you need to put in smooth weight functions everywhere. But that is, is basically method and it's some, it's actually a little bit easier than the most standard variants of the hardy little wood method because there's only one major arc about the point zero. There's no major arcs about other rational points in this in this very. So I won't say any more about the details of that but I want to tell you just to finish a couple of things about some slightly interesting ingredients that go into that application. So to make it work. You need a little bit of random matrix theory first of all which was quite surprising to me. And then to get this very tiny exceptional probability. There's something that I call a combinatorial amplification argument. So it doesn't drop straight out of the hardy little wood method you first apply the hardy little wood method and then you do some some kind of an amplification to get a very, very strong error term. So just one slide on each of those two things to finish off. So first the random matrix theory. I needed a lemma like this, not for four by four matrices and need it for n by m matrices, but four by four was the biggest I could go. So random symmetric matrix that's constrained to have ones on the diagonal, all of the other matrices are between all of the other entries are random minus one up to one. Then the claim is that the probability that the determinant of such a matrix isn't most delta is bounded by a constant times delta. Okay. So there are two things here. The first issue is I've got ones down the diagonal which is a bit nonstandard. So I also need this for n by n matrices, and I need it again with just Delta and I need to make sure that the dependence on n isn't absolutely terrible. And I also need it not just with ones on the diagonal but with other fixed diagonal entries but let's let's gloss over that. So the two ideas in the proof so first of all get rid of this annoying feature that the diagonal entries have to be fixed. The idea for that is what I call amplifying the diagonal. So I think this is a fairly standard sort of procedure. If you did have a big set of matrices like this, or with small determinant, then you could make a big set of another, an even bigger set of matrices just by multiplying through by diagonal matrices on the left and right. So diagonal matrix D multiplied through on the left and multiplied through on the right by D, that will still have small determinant. But now it won't have fixed diagonal entries that look more like just an arbitrary symmetric random matrix. So if the problem is if there were many of these matrices with small determinant, there'd be many symmetric matrices. So that gets rid of the fixed diagonal entries issue. So that's not the issue of how to prove that a symmetric random matrix with uniform entries doesn't have large determinant. And I thought this would be something that just immediately you could look up in the literature that I wasn't able to find it. Well, one thing you can do is you can compare this uniform random matrix model to one of the standard models like the GOE ensemble. So this is the random matrix ensemble with symmetric matrices with Gaussian entries. And there there's an explicit formula for the joint eigenvalue density. And of course the determinant being at most delta is just the same as the product of the eigenvalues being at most delta. And what you have then is just a computation to do with that joint eigenvalue. And then you can kind of embed this uniform random matrix model inside the the GOE model in some fairly simple way. So I think probably the reason this isn't in the literature as far as I know is that usually in random matrix theory you don't consider this this particular limit you'd be more interested in the large end limit. Whereas for me and is fixed like for in this picture. And I'm interested in the sort of small determinant probability. Okay, so just the last couple of minutes I want to tell you something about this combinatorial amplification trick so just give a very rough character of how that works. So let's suppose I've got forms in seven variables could take quadratic forms in seven variables. I want to fix the diagonal terms, a one one up to a seven seven and choose the off diagonal terms a i j independently at random. I want to show that the quadratic form with these coefficients doesn't have any large gaps. And what I can do is I can restrict to various subsets of the coefficients. So I've got a Q is the quadratic quadratic form in seven variables. And I've written down here a green form which comes from setting t t one t three t four t five all equal to zero. So just considering t two t six and t seven as variables. I've got a red form where I've set t one t four t six and t seven to be zero. And then a blue form I've restricted t one t two t three and t six to be zero so these are various different restrictions of Q. So if one of these restrictions has small gaps, then the big quadratic form Q in seven variables also has small gaps it's a hereditary property. So I just need one of those events to hold one of the events that either the green form or the red form or the blue form has no large gaps. So these range over a discrete set of points in a box. But these forms those, those conditions are independent. And the reason they're independent is if you draw those off diagonal coefficients. Well they are disjoint triangles in this. This copy of the complete graph on seven verse seat. So actually this is a well known picture this is a decomposition of the final. This is a plane basically into lines. So it's a perfect decomposition of K seven complete graph on seven verses into disjoint triangles. But now this this sort of shows why it was important to be able to fix diagonal terms, because the diagonal terms are definitely not independent so a to two appears here and here. So these triangles that edge just joins and not vertex destroy. So only get this independence if you're allowed to fix the diagonal terms and only vary the off diagonal terms. So that's why it was necessary to deal with that weird random matrix problem on the previous slide. Okay, so if just one of those events holds, then the form Q in seven variables is dense on 0.9 up to 1.1. So you can massively improve the exceptional properties of the probability that the green form is not dense is something you just get to multiply. Okay, so that's all I want to say. Thank you for listening.


Hello everybody, E here. Welcome back to another book haul video. This is my July book haul, and we got 27 books to get through. It's kind of ridiculous. I'm going to do at least one bulk thing. In fact, that's what I'm going to start off with. We're just going to jump into it because we're probably going to be here for a while. So grab your snacks. Let's talk about why I got these books. I almost dropped this whole stack of books. So let's see here. Right off the bat. I can't wait for the comment section. I thought you were done with Coombs. I got several books, paperbacks that I don't have for Dean Coombs. They were only $0.25 a piece. I spent $1.25 on it. Didn't see any reason to, not to. Because Mr. Murder, the Frankenstein series, Velocity, and Icebound, I don't have any of them in paperback, but I do have Mr. Murder in hard cover. I don't think I have any of the other ones. So Mr. Murder is the only one that I have. I have it in hard cover. Let's see here. Yeah, I have it in the small hard covers that I really don't like. I like the full-size hard covers, but I tend to collect these and then sell the collections on either eBay or to friends anyways. But I keep the paperbacks most of time unless I have built whole collections of Dean Coombs three times and then sold them. I've done that with Richard Lehman twice. So especially around here where people just give their books away. I find them at the library at thrift stores for a quarter of piece and I just build up collections and then sell them. Yeah, you know, why not? Anyways, let me go ahead and put these over here. But I've read all these books, all these Coombs books. I don't think I liked any of them, to be honest with you. I really hated the Frankenstein series, but it had very little to do with the content of the books. When we get to that series in my Coombs reread, I'll discuss this more. But the first three books, Dean Coombs didn't write alone. The first three books he wrote with Kevin J. Anderson, several different people, well three different people total, one for each book. And then he just took their names off the books when he reprinted them. I don't know why. He says that he's always worked better alone, but he worked with those people and then he took their names off the books. So I'm a bit confused here. Next up we're going into some terrific, terrific finds, getting away from the Dean Coombs nonsense. Clive Barker's The Books of Blood found this for a quarter of this is volume two. So that'll help with my not reread, but my read through, because I've only been able to finish his short stories. I do have several of his novels, but I've never even been able to get into them. I've gotten about, I don't think I've gotten farther than 50 pages in any one of his novels, other than The Thief of Allways. And it's one of my favorite books. I easily put it in my top 100 books of all time. It's such a fun book, but I don't like his adult novels. So I don't know, but he's our greatest living storyteller. Then next, I can't believe I found this y'all. I don't know if you remember or not, but I talked about this, I think in a Stephen King video, it might have been one of these book haul videos, but I finally found from the Borderlands. Yeah, stories of terror, whatever. But this one has a story in it by, let's see here. It's the, yeah, Rami Temporalis by Gary A. Bronbeck. It's about this guy collecting the pieces of God's face. Such a weird story, but it's one of my favorite short stories of all time. And of course, it's got some stuff by, I think, Bentley Little, John Ferris, Brian Freeman of Cemetery Dance, no less. Let's see here. I think Whitley Shriver's in here. Bentley Little, let's see here. I can't, Bev Vincent, Stephen King. I thought, yeah, Whitley Shriver, Striever? I don't know how to pronounce it. Father Bob and Bobby. But yeah, these two are probably the coolest things I found for a quarter a piece at the thrift store. Sometimes I'll find them at the library for a quarter a piece. This is a thrift store hall. We have a place in town called Hope in Action. It's a Christian bookstore and they usually have amazing horror collection. I've bought most of their stuff. So now it's just like Bibles and Joyce, whatever her name is, Joyce Meyer books. Oh yeah, and they got, they probably, they have, well, not probably, they have a whole shelf of Sarah Palin's Gone Road. Yeah. All right, so next up we have Thomas Harris's, you know, you'll be able to see that Black Sunday. It's the last one I need for my paperback collection of his stuff. I own all of his books and paperback. I do have Cardi Mora in hardcover back here. And I'll get to that eventually, even though everybody's like, no, do it. Run away from the book or show back. I'm gonna find out for myself, y'all. I'm gonna find out for myself like I always do. But yeah, so I've never actually read this one. I thought I did, but the synopsis doesn't sound familiar. So, and I think they made a movie of this. If they did, let me know down there in the doobly-doo. So, all right, last of the paperbacks, well, not last of the paperbacks, the last of the mass market paperbacks, is a gift from my friend, Terry Nevermore by Harold Schechter. Is it Schechter? It's Schechter. Yeah, it's got a Raven, Nevermore. Yeah, it's got a Raven on there. That's some blood, blood dripping.  If you look at the blood on the cover, you could tell that the blood was just like photoshopped on top of it. There's like, there's weight. It's like it's hovering on the actual cover itself and not on the ground. It's pretty bad. The bird's pretty badly photoshopped in there as well. Okay, next we're gonna get to, let's see here, I don't know what to do next. I got 27 books today, y'all. I don't know what to do. So we're just gonna go, I should have probably organized these by gifts also, because I just got everything just kind of all slapped together. But this was a gift from my friend, JB Taylor. He runs a channel called JB Reviews, I believe. I think it was a recent, shoot, what was it? Or Taylor, Taylor reads the reviews. He's changed his name at least once, but he sent me this Kafka on the shore. I own this, but I don't own this copy, this version of it. And I love these versions just as much as I love the new rainbow ink ones. Those I think are just beautiful on the shelf. My book tour, I think my third book tour, bookshelf tour, you can see my collection of the rainbow ones. Those look gorgeous on the shelf, but I do love these ones. They have really boring spines, but the covers are very, very nice. So there's Kafka on the shore. This will probably end up being my reading copy, because my rainbow one is in pristine condition. Next up, we have a book that has a funny little story behind it, I think. Not the book itself, but how I came across it. No, Invisible Man, by Ralph Ellison. This is a book about race. I believe it's a literary novel about race. I thought this was The Invisible Man. I didn't realize that H.G. Wells wrote The Invisible Man, and Ralph Ellison wrote Invisible Man. So the difference between the two books is the. H.G. Wells has the Invisible Man. But I bought this and I was like, wow, I didn't think H.G. Wells wrote books this thick, and he doesn't, or didn't. I don't know what this is. It's just a receipt. Okay, so that one's going to go over there. Next one is a book I bought that I think I own a copy of. When I got home, it says on Goodreads that I own it, but Winterwood by Patrick McCabe. So if I have two copies of this book, whatever. It was only a quarter at the library. By the way, Invisible Man was only a quarter at the library. Also, no wonder I couldn't pass it up, right? I mean, come on now. Come on. Let's be honest. Next up is one I have to show you the spine of because it didn't come with a dust jacket. I think this is volume. I'm not sure. It doesn't actually tell you in here. That's unfortunate, but I believe that this is volume 35 of Weird Tales. The dust jacket for this version is amazing. I believe it's the one from volume 35 because I went in and typed in the highest BN. If you want to look it up, it is ISBN<PHONE_NUMBER>. I will probably read everything in here except for the I will try. I'm not going to force myself to read anything, but I will try to read everything in here except for the Lovecraft, because as you guys know, I don't like Lovecraft. Then forget all the other BS with how terrible he was as a human being. I just don't like the way the dude writes. But there's some stuff in here from authors that I have enjoyed in the past, like Clark Ashton Smith. See here, there's Ray Bradbury has Let's Play Poison. I love Ray Bradbury, by the way. Let's see here. There was someone else that I got happy about, Theodore Sturgeon. That's another one. So volume, I believe 35 or 36, it's 30-something of Weird Tales. I thought I had stumbled across something that was rare, but no, that obviously they had several different volumes over the course of time. I never actually read, I never actually got a chance to read any of the Weird Tales books when they were active. They came back, I think. Well, of course, I don't even think I was born while they were active. Was it, they closed in the 70s? I can't remember when it was. It tells you in the, I actually read the Introduce the Forward. Yes. Yeah, it ceased publication in 1954, but then they brought it back for the internet age, or somebody tried to. And I'm not sure if they closed again. If you guys are more knowledgeable about Weird Tales than I am, please let me know down there in the doobly-doo. Also, again, I'm not going to sit here and read all the synopsises and everything. I'm just going to tell you why I bought all these, because I don't like watching someone just sitting reading synopsises for 30 minutes to an hour long. I would rather tell you. And if you're interested in the book, you can hop over to Amazon and check out the description. So, let's see here. Next one, next two, I'm going to go ahead and throw up two, three, three. No, I'll just do two. There is Faceless Killers by Henning Mankel. Yeah. It's the Kurt Volander series. So there's that one.  It's so gorgeous, man. It's beautiful. Oh, also, I didn't show in the invite. These are gifts from my friend Terry, by the way. But the author's photo is right here, too. I usually show the author's photo. But, uh, anyhoo. And then there's him looking very much like John Irving. I don't know if it's going to show it or not. Anyhoo. But, uh, is it going to focus? It's focused. Okay. I was worried. Oh, worry, boy. I'm patching me down there in the comments. I'm up. You're out of focus. I know, buddy. I know. And I know I can cut and then start over again, but it's a lot of fun. Just, you know, if I can make it entertaining, while you're waiting, I can make it entertaining. So, all right. And then the next one is a gift from my friend Terry.  Uh, I'm going to probably get to this one before the end of the year. Um, I've been putting it off far too long, and I really want to try his, uh, his longer work because I read some of his short stories.  All the stuff from Terry in this pile is for my birthday. Uh, let's see here. Next up are two books. I'm just going to put, uh, Caroline Kepnes, uh, was on Twitter and we were talking back and forth with Amina Oktar and, uh, Araminta Hall and all of us were talking about books like their books. Like hashtag fashion victim, which is, uh, Amina's book. Araminta's book is, uh, uh, our kind of cruelty. And of course, uh, Caroline's books are You, Hidden Bodies, and Providence. I love all three of those books. They're just fantastic books and they got to talking about their inspirations and books like their books. Caroline Kepnes brought up, uh, of course the, no, no, no. Araminta and Amina both brought up the Patricia Hightower. God, I hope I got that right. High Snow. High Smith? I, I can't remember. The Thomas Ripley series. I didn't care much for that. I know, I know. Rage at Me down there in the doobly-doo. Um, but I wanted to know, are there any other books that are like your guys' books? And Caroline, uh, brought up Perfect Days by Raphael Montes and then The Beloveds by Marine Lindy. And when Caroline Kepnes says buy something, man, I am, I'm definitely a follower in that regard. So yeah, both these books. Also, this cover is fantastic. That bleeding palm tree, man.  Um, but yeah, I'm looking forward to this one. I'm looking forward to both of these. These are probably going to be winter reads for me, uh, just because I like that kind of thing during the winter. And next up is a The Turtle Boy. This is just Peregrine's Tale. This is the short story. It's technically part four in the Timothy Quinn series. Uh, and my friend Nick was nice enough to send me this. This is absolutely amazing. It is a signed copy, uh, limited to 550 copies. Keeneland Patrick Burke, and I think the, uh, is at the, who's the other signature? Is it the artist? I'm guessing it's got to be the artist, right? Hey, let me see. Alan M. Clark. Ah, sorry, I had a mosquito on my finger. What the actual, anyways, uh, Alan M. Clark has illustrated the writing of such office as Ray Bradbury, Robert Block, Joe Arland, Bill Stephen King, George Orwell, and so on and so forth. So yeah. Um, so Arthur M. Clark did this. Keeneland Patrick Burke wrote it. Um, so all the art is Arthur's. I don't know Arthur. Why am I calling him by his first name? I don't know. But yeah, that was super nice of Nick to send, send over. Uh, let's see here. What else we got? All right. The rest, no, one, one more paperback, uh, trade paperback. This one I screwed up with. This is the Lizard King, the essential Jim Morrison. This is not, uh, his poetry. I thought it was his poetry. I should have flipped through it and find out. But this is actually interviews and, uh, some, maybe some short fiction, some biography stuff or whatever. So yeah. Um, not too thrilled with this one. I'll probably just end up either giving it away or selling it. If anybody wants it, uh, comment down there in the doobly-doo and we'll work something out. I, um, okay. Next up, another one that Terry sent me for my birthday was Wiley Cash's, uh, This Dark Road to Mercy. It's a library copy. I will probably, uh, take all the stuff off and clean it up. Oh, it's got deckled edges.  I don't know why, because I like flipping through a book to look ahead and it is really hard to do, not to look ahead, the, to see how much longer I have on a chapter, that kind of thing. Um, and you can't really do that with deckled edges, but I, I love the aesthetic of them, which again is weird because usually you're putting the book in to wear the spine. Well, not usually all the time. So you don't even really see them, but I like the feel of them. I like to look at them. I don't know. Am I weird? Yeah, I'm weird. I'm weird. Give it up, y'all. Next up is a gift from a friend. Once again, JB Taylor sent me his copy of Vox because he hated it. I love the idea behind this. Um, so I'm excited to read it and try it. Um, who knows? It is blurred by Lee Child. I wasn't expecting that. Um, I'm not a huge fan of Lee Child or Vanity Fair or The Daily Beast or, and I don't know the other people who blurred this book. So now I'm kind of worried. I didn't realize that. Um, me and JB tend to have the same, the same taste. Uh, he likes Haruki Minakami. He likes Stephen King. Um, but we do tend to differ on some things. Like we liked, uh, Colson Whitehead is the Underground Railroad. I can't remember what else, but we tend to normally fall on the same category, but the things that he didn't like about this one, I think I might actually appreciate. So yeah, I'm gonna give this one a shot. This is another one I'm going to read, uh, pretty soon. I also have the audio book I forgot I got. Next up, just for you guys out there that like to scream at me about Dean Coons. I got 77 Shadow Street in hardcover because it was a quarter at the library. It's the same day that I picked up the, the paperbacks. So yeah, I mean this one, from what I understand, this is one of his worst books of all time. Uh, damn near all of the Coons fans that I talked to, uh, the few that I do talk to hated this one. Um, in fact, uh, I think there are two comments about how terrible this book is on my Phantoms review. I believe I could be wrong. Um, but yeah, so I have no, I don't think I've ever read this one. This is after I gave up on him. Um, I tried Ashley Bell, but uh, yeah, and especially this is one of his thicker, thicker books. Uh, in hardcover, it's, uh, 445, 46 pages, something like that. Uh, yeah, 451 pages, and I just, I wasn't gonna spend that much time with it. Um, I tried Ashley Bell, which was another one. Uh, I ended up giving up on Ashley Bell within two pages, but we're doing the Coons chronological reread of the books that I can get my hands on. So yeah, my buddy Cody Tidwell is actually doing all of them, even all of the old, out of print books. So I don't know if he's planning a review series or whatever, but I will definitely link you all and, you know, you guys can check him out. I don't, he doesn't do videos, but you can definitely check out his, uh, his written reviews though then. Next up is another one that I got at the library four quarter, uh, for my, just my Book of the Month Club collection, and that's an anonymous girl.   If I'm wrong, let me know down there in the doobly-doo, uh, if this is just another generic, you know, uh, domestic thriller. Um, I do like Lisa Jewel, but she's about the only one I've seen do the domestic thing, right? Um, I don't agree that Caroline Ketnes is a domestic thriller, um, because it doesn't have to do with domesticity, domesticity, whatever. I think you is a horror novel. I honestly believe you is a horror novel, because it terrified me anyways. But anyways, so, uh, what is this thing? What is this? I mean, oh, I do love Nathan Ripley, uh, his, uh, Your Life is Mine. That one isn't really a domestic thriller, but, uh, what is it? Find You in the Dark? Is that the name of it? Where is it? Yeah. Find You in the Dark was a terrific domestic thriller. Um, then again, it wasn't really domestic, because the point of it wasn't really the family, but it was the family, it wasn't care. I don't know, what is a domestic thriller? Isn't it just like when the wife or the husband, they're married and they're, what's a domestic thriller?  So we're going to talk about The Gifted School, uh, by Bruce Halsons, sure. This book, uh, was sent to me, this is the last one that was sent to me for my birthday by Terry. This is probably the only one that I might not ever read. Um, I don't know, it sounds interesting, it's just not my kind of thing, it's more like a contemporary experience. If nothing else is going to look fantastic in my Book of the Month Club collection. Um, but, uh, I, all the stuff that Terry sent me, of course I appreciate, but this, I don't, I'm not sure that this one is my jam, you know what I'm talking about. Anywho, next up are the, the two that I bought, I guess brand new, one of them that, one of them was on clearance, and it might actually make some people upset, because Cemetery Dance said something, they said that their, that their first, uh, their first editions sold out. Um, I'm wondering, I got a first edition of Flight or Fright on clearance at Bar, not Books a Million, um, for $7, so that's the only way I was going to get it. Um, I didn't realize it had such a, such a great, uh, table of contents though, such a great collection of authors. Got Joe Hill, uh, Cody Goodfellow, um, let's see here, uh, Ray Bradbury, Roald Dahl, Stephen King, James Dickey, man. Um, yeah, so I, do they count the stuff that they send to the, is that considered sold out? If they, I mean, that doesn't make any sense, because, you know, if they were to send all the copies of like a Stephen King book out to Books a Million, then it just seems, I don't know, it seems fishy. Y'all let me know how that works as far as the publishers are concerned, because they said this joke, the first printing, first edition first printing sold out, but yet there's a whole, there's whole stacks of these up at Books a Million. So I'm wondering if maybe, I don't know, how does that work? Y'all let me know in the noobly-doo because I'm kind of, I'm kind of sketched out. Anyways, last one is one that, uh, Don Wilson has been talking about for months, so I went ahead and grabbed it. Uh, the premise and the idea is great, I think, um, but I am worried that it's going to end up being a generic thriller. And that is The Chain by Adrian McKinty. I think he's a Irish, and he wrote an American thriller. Um, the only reason I bring that up is because they, uh, some of the blurbs have mentioned, um, how well he wrote an American novel, a novel, uh, set in America, an American voice novel, yet he's not American. So, and that's another reason why I picked it up, because it's not an American author.  Don Winslow, Stephen King, those guys are just, are different in that aspect. Um, but, you know, I'm talking about, like, the James Patterson's, the Dan Browns, the Clive Custlers, you know, all that stuff. Uh, usually when an international author makes it over here, like Yonesba, their books are at least decent. You can finish them and enjoy them, you know, anywhere from three to five stars. I've never read an international novel or a translated novel. I know this isn't translated, but, um, I've never read an international or translated novel that I didn't at least give three stars, I think. If someone wants to prove me wrong, definitely jump over there to the Goodreads and find one. But, yeah, so the chain, um, it's got blurbs on it. People were picking on me for this. I said that Stephen King, I don't think, he's made me blur two good books out of the hundreds of books that he's blurbed, and there's a blurb on the front of this from Stephen King. I didn't know about it before I bought it. Uh, even though I love Stephen King and I'm a huge fanboy, I think he's got terrible taste in movies and books. That's how I feel. So, Don Winslow said, this book is jogged for parents, and Stephen King said, this nightmare story is incredibly propulsive and original. You won't shake it for a long time. What the book is about is, I believe, someone's child is kidnapped, and the only way they can get the kid back is to kidnap someone else's child, and that's the chain.  Uh, excuse me, but it's like a chain stealing. Well, chain kidnap, I don't know. Anyways, it's also got blurbs from, uh, nobody that I really care to mention. Oh, make Gardner, um, diabolically gripping, hang on tight, because once you start this book, you can't stop. Like the characters in the story, you'll be caught in the chain. Um, unfortunately, Amazon sent this, and there's scratches.  They got, there's rubs and scratches all, I don't know that you guys can see this, but there's rubs and scratches. Maybe if I get in the light, no, I still can't see it. There's dents and scratches and all different kinds of crap, um, on this. And this is the last new book I'm buying from Amazon. I'm done with them. I'm just not going to screw with them anymore. So I have to find another place to buy my new books, um, and I don't shop at books a million other than clearance, and I don't shop at Barnes and Nobles, because they just, I mean, it's insane prices. Full price, MSRP, or whatever you want to call it. It's the, it's the price in here is what it is. Um, and I buy so many books, I just can't afford the full cost of a book. That's the only reason why I shop on Amazon. So if you guys have any, let's see here, if you guys have any places that maybe I haven't heard of, just list them, even if I have heard of them, just list them down below, because I'm sure other people want to try and get away from Amazon for new content also. Um, and I don't, I don't read ebooks, so really Amazon is kind of pointless at this time. Um, in fact, we're going to cancel Prime come next month after we finish, uh, the boys, um, that we're watching on, well, Amazon Prime. But, uh, which is extremely good, by the way. So that's my July book haul. We've been here for at least 25 minutes, I think. Um, so I end all of these the same way, and I want to end this one once again the same way. Book hauls are some of my favorite content. In fact, I would go as far to say they are my favorite content on this platform. So if you guys have any book haul videos, please link them down below. It's a call to arms. Link all your book hauls down below. Another thing is, if you don't have a video, just list what books you have. I mean, there's no better way to get to know another reader than to find out what kind of books they're buying. That way I can judge, like, hey, why did you get this one? You know, this one sounds interesting. You know, this, that, or the other. We can have a conversation. It's one of the best ways to have, to start a conversation with a group of readers, is what kind of books do you buy? What did you buy this month? What had you in the mood for this one? Anyways, I say this at the end of every book haul video, but until next time, I have been E, you have been U. This has been another book haul video. I'll talk to you guys later. Bye-bye!


Highbridge, a division of recorded books, presents the goodness paradox, the strange relationship between peace and violence and human evolution by Richard Rangham, read by Michael Page. Preface At the start of my career I would have been surprised to learn that fifty years later I would be publishing a book about humans. In the 1970s I was privileged to be a graduate student working in Jane Goodall's research project on chimpanzees in Tanzania. Spending whole days trailing individual apes in a natural habitat was a joy. All that I wanted to do was study animal behaviour, and in 1987 I launched my own study of wild chimpanzees in Uganda's Kibale National Park. My bucolic research was disturbed, however, by discoveries that were too intriguing to ignore. Chimpanzees exhibited occasional episodes of exceptional violence. To shed an evolutionary light on this behaviour, I compared chimpanzees with their sister species, bonobos. In the 1990s research on bonobos was beginning in earnest. Chimpanzees and bonobos were proving to be an extraordinary duo, bonobos being much more peaceful than the relatively aggressive chimpanzees. In various collaborations that I describe in this book, but most particularly with Brian Hare and Victoria Wobber, my colleagues and I concluded that bonobos had diverged from a chimpanzee-like ancestor by a process that was strongly akin to domestication. We call the process self-domestication, and since human behaviour has often been considered similar to the behaviour of domesticated animals, the insights from bonobos suggested lessons for human evolution. The key fact about humans is that within our social communities we have a low propensity to fight. Compared to most wild mammals we are very tolerant. I was acutely aware, however, that even if humans are in some ways notably unreactive, in other ways we are a very aggressive species. In 1996 in a book called Demonic Males, Apes and the Origins of Human Violence, Dale Peterson and I described evolutionary explanations for similarities in aggression between chimpanzees and humans. The pervasiveness of violence in human society is inescapable, and the evolutionary theories explaining it seem sound. So how could our domesticated qualities and our capacity for terrible violence be reconciled? For the next twenty years or so I grappled with this question. The resolution that I describe in this book is that our social tolerance and our aggressiveness are not the opposites that at first they appear to be.