Happy holidays, Lone Fox family. I cannot wait for today's video. We are decorating for Christmas time. So this is the first time this home is seeing holiday decor. And I will say over the past couple of years, I have acquired quite a bit of Christmas decor and abundance of it. And I'm kind of hard to part with holiday decor for some reason. So because of that, we have a lot to work with. I also do have a nine foot tree, which I didn't use in the last couple of apartments because I got it when I lived in downtown years back. I've saved it for many years now. So we're gonna bring that up, put it in the main living room. It's flocked and I'm so excited to see that in here. I also got a new sofa in the main living room. I'll give you guys a little look. I actually picked this one up from popup home. It's a vintage shop in Los Angeles. Honestly, I followed them for years and I always love what they post. And I just saw this one pop up on their Instagram and I just kept thinking about it for like two weeks. I couldn't get it out of my head. Finally I got it and it's so comfortable. I love it. It's like an Italian 80s leather-rushed situation. But yeah, we have some decorating to do. Editing drew popping in quickly because I just realized that our Black Friday sale started yesterday and this video is being posted tomorrow. Take advantage of our Black Friday sale. It's the biggest sale of the year, 20% off site-wide. So everything you see in this video, like all the Christmas decor, everything is 20% off. And in case you guys didn't know, I have an app on iPhone, iOS, and also on Android and it makes the checkout process, the shopping process so easy and simple. I just, I love shopping through the app. It's really easy. I finally figured out my new camera and I really like how like wide of a frame I can get. It's so cool. And I need to kind of figure out where we're gonna put the tree. Now I do definitely want it on this end just because you enter from where the camera's at and it's kind of going this way. I pulled the chairs apart because I have an idea. I found this tripod Christmas tree base on Facebook Marketplace for $25. And I just thought it was so cool. So I picked it up. I think I'm gonna put the tree in it this year. I was thinking that we can either have it go directly in front of the door. We can have it over on this side or we can have it over on that side. So there's like three options as to where the tree can go. I am actually going to do an Instagram poll right now and ask where you guys think the tree should go. So if you're not following me, Lone Fox home right here. All right, so we have three options here.  I think for now I'm going to put it in between these chairs because we can move it. So it can be moved around. But we're gonna do the mantle which this mantle is gonna be a little different than downstairs because there is not a mantle here as you can see because they removed it and brought it back to more of the original style of this fireplace. But I do have a couple of garlands from last year and these are packaged and for some reason they smell like barbecue. Full on like kind of an odd scent for a garland. I do have a nail on either side from decorating for the full time. So for this fireplace I started off with just two garlands and I draped those across the front and then added a strand of bells and this was getting me at a good starting point but I knew I wanted to make it look like there was kind of like some drape on either the left and right side of this as well because I feel like it highlights the frontal look of the fireplace. So I added these bell clusters and then I had some ribbon from last year. All of this that I'm using actually is from the last couple of years. So I keep every little ribbon scrap possible. These were on the tree last year kind of wrapped around it but I kept them and I'm turning them into bows on the garland and then wrapping a few strands that I had. I only had like four pieces of this ribbon total. So I was excited that I was able to get it usable on this mantle area. And a little detail I have to highlight is this curly ribbon here. Now it was just ribbon on a roll but when you pull it off and just kind of like place it wherever and don't pull on it it stays curled like that which is so cute. I popped in some blueberry garland which I love. I got this from Anthro this year which I feel like so unexpected for Christmas time and that's the mantle. I moved the tree to the center and I kind of am liking it. I asked on the pole and a lot of you guys said to move it to the center and I don't know why. I would always think to put it on the left side over here but that's why I love asking. So it's in the center. I opened up the doors as well just to let a little breeze in. Look at this background. It's looking so good in here. If you've been around on the Lone Fox channel for a bit you would know that I have an online store. It is lonefox.com and every year we do like a big holiday collection and it's normally about 100 items but this year it is almost 300 items that I literally sourced myself. I went to market at the beginning of the year I found everything for the shop and we're gonna be decorating my trees in my home with a bunch of decor from the Lone Fox shop. So definitely if you're looking or in the market for some holiday decor this year check out my website. I will link it at the top of the description box. I have just a small assortment of some of the ornaments that we are offering. So I'm gonna be adding these like gilded brass ornaments. I think are so cool. We have these plaid ones that I love. I've just been so into plaid lately and they come in three different colors. We also have so many different like little specialty ornaments such as little owls. There's cocktail glasses. There's like butter sticks, croissants, cheese, passports. These ones are my favorite that we're adding to this tree in particular. They actually come in a set of four. So you get these three and then this one right here and they're paper mache. But how great is the pattern on these? I think they're so fun. Like they kind of give off like whimsical circus vibes almost. This year I got these as well. I thought they'd be fun. They're dried like ferns and little flowers inside of these glass, very like delicate bubbles. Every single item up in this little nook up here is from the site. There's a bunch guys, even gift wrapping. Also got to do a little holiday promo, you know? So check it out. Now I'm going to try to find a very similar tree to this one. It really is such a beautiful color. It kind of has a sagey kind of very light tone to it which I love. And whenever I place ornaments on a tree there's no rhyme or reason to this. Just have fun with it. Marie was helping me out. We do this every single year. We have lived together believe it or not for nine years now. So yeah, this is kind of a tradition. We are going to be putting garland around the arch into my office area which I'm so excited about. And I do have kind of a spoiler alert. This office area might be changing in the future. Just a little bit. I have ideas, I have ideas, I have them. So Justin went down to the flower district this morning and picked up some garland while I was editing. And the flower district in Los Angeles is in downtown. We got 30 feet of garland, real garland. There's three types of greenery in there. I don't remember any of them except for Cedar. Except for Cedar. Cedar, noble, noble kind. It was $6 a foot. So if you're curious on the price, it was $180 was what we spent on the garland. But that's really not too bad. We get 30 feet of it. And it's handmade. Oh, this is so beautiful. It really is. It smells so nice. That's 20 feet long. We ended up cutting it because the direction flows in one way. So now we're gonna swap it and then have it flow the opposite way. So I will say, if you're gonna add real garland, you probably will have to use a couple of nails, which we had to do.  So you can probably just go in with paint after the season's over with to touch them up. I also ended up moving the tree over to the right side to see how I'd like it for a bit. And then this doorway here, which actually leads into the living room got kind of the bad selection of garland, like the little end pieces, but I kind of made it cute still. I feel like it looks folky. There's like a little crown on the top. And yeah, it's like having a bad hair day, but kind of cute still, you know? And then these ornament stands, these are so fun. I found them from Anthropology this year. It's to display little specialty ornaments, or if you wanted to kind of put an ornament on a shelf, this is a great way to display that. I love this. Kept the coffee table styling pretty simple. Just added a tray, a candle, some of these reindeers from the shop. And whenever I say the shop, I'm referring to my store online. So if I say that, just know it's from the shop. And this little bin here, the little like corn box, I love that. It's also from the store, it's sadly sold out, but it has a remote and everything inside of it. And that is the coffee table, super simple. And I popped a throw pillow on the sofa, but look how great it's looking. It's Sunday today, and I'm doing some laundry finally, because I just got the laundry room makeover up. And if you have yet to see that video, you have got to check it out. It is such a great laundry room. I love the way it looks. And so I'm washing my bedding. You can see my Helix mattress, which happens to be the sponsor of today's video. I just love my Helix mattress. Like look how thick and juicy this mattress is. It really is the most comfortable mattress ever. Had it for years now. I feel like she's just broken in. Like I can have her for so many more years. I absolutely love my Helix mattress. I mean, there's really nothing to dislike about Helix. First thing the mattress has come shipped in a box like this. I find it so convenient, especially if you have to move it to a room, like who wants to move a mattress around their entire home. This box is so much easier. You get to unroll it just like this and it puffs up. It probably grows like 10 times in size. It is unreal how they're able to compact a mattress in a box like this. And on top of an incredible product, they have free shipping within the US, a 10 year warranty and flexible payment plans and financing options, because a mattress is definitely an investment, but Helix makes it so easy to make sure that you have quality sleep that won't break the bank. And I kind of feel like I have been growing up with my Helix mattress over the years. It gets more comfortable as you sleep in it, I swear.  Plus you get a 100 day sleep trial, meaning you can try it out for 100 days. And if you don't love it, they will come back and pick it up. You do not have to put it back in the box, which is incredible to make sure that you love your mattress. So definitely take advantage of Helix's incredible Cyber Week offerings. This is our biggest sale of the entire year.  And this is for a limited time only. So make sure to take advantage. You can visit Helix Sleep site for more details. I also got a new bed frame, which has a drawer on the bottom of it. How cool is that? I'm gonna share more of that when we get to my bedroom makeover, which hopefully is gonna be the next space that we're working on. But I actually want to put a Christmas tree in the corner over here. And over the years, I have kind of collected a couple of trees, especially working with a few brands during the holiday seasons. Sometimes they want you to use their tree in particular so they send it over. So I've collected a few and I have an additional Christmas tree. This one I actually had in my living room last year as the main one. This is a nobillus, or nobill, nobillus, nobillus fir, pre-lit tree. And this is from Terrain and Anthropology. It's a seven foot tree. I also want to share this hack with you. So you can purchase these woven tree collars and they're always marketed like this, like kind of as a collar that goes under the tree. I just flip them, which I don't know why I never thought to do this the first like two years I had it and it turns it into a basket. And I feel like it's something that you just might not think to do because a lot of times tree collars are always this way. They're always marketed in like this kind of like upward cone shape. But flip your tree collar and you have more of a basket shape for your tree, which I think looks so cute. To decorate, I want to go a little more simple on this tree because the other two I did a lot on. And so for this one, we actually have these stunning, like beyond gorgeous silk ribbon in the store. And you get all of this guys. Like you get all of this that you see. I think it's like 75 yards or something like that. It's an insane amount, $15. And it comes in a bunch of different colors, but look how pretty. I think it's like very elevated for the holidays and it's perfect for gift wrapping.  You get it kind of like wound up in this little ball bundle like this. Go and do it and it just pops open. Just so into red right now. I kind of want to do the burgundy on this tree. I feel like it's just so rich in this color. It's the darker red. We have two different red options.  So why not put an ornament hook on your tied bow and then you can take them on and off. The thing I love is that you get a whole bunch of this ribbon too. So you can do very kind of like lazy bows, which is what I like, the very casual, more floppy, like elongated tails. Like this is the kind of bow I'm looking for. You know, like this is a bow I like. Your ornament hook added to our tree branch. Okay, that's actually cute. So I just kind of cut a long, generous amount. Very French kind of like pretty ornament hook slipped through. I added a bunch of bows to the tree and I'm loving it. I still have this much ribbon left from that $15 thing of recycled silk ribbon. So this, you guys, if you want some, get it because it's been going quick on the site. I definitely do still wanna add some of these little silver ornaments as well. So I'm just gonna pop some of these on. And our last step is to just put a blanket in our basket at the bottom to cover the base. So I'm about to start moving around the furniture in the lobby area. One of my absolute favorite features of the slipping room is the egg-shaped window right here. I love it so much and it's definitely the focal of the house. Like when you see this house, it is the first thing you notice on it. And in this window, I want there to live a Christmas tree. So we're going to start moving some of the furniture. This incredible sofa is modular from Castlery, which I love, so we can kind of move it around, piece it together. I kind of wanna do something symmetrical on either side. Here is the tree. It's so cute in front of the window. I seem to find the right light setting. I feel so great in here actually, like you can actually nice conversational vibes. Oh, I still do have this really cute Lulu in Georgia chair, which I can kind of have facing inwards. I had a second one of these chairs, which I haven't got to use yet. So I just put them next to each other here. Apologies for the split screen filming of this mantle. So I actually ended up filming this on my phone for TikTok and Instagram. And after filming it, which it took me probably two to three hours to do this mantle, I really put a lot into this one, layering up some garlands first. But after I did it, I just didn't have it in me to take all of it down and have to redo all of it horizontally because everything was just so perfectly placed. And after that many hours, I was just like, I cannot take this down. So I'm sharing the footage I filmed on my phone, which I first layered in some garlands, as you can see here. And I'm kind of framing out my fireplace because it has a niche on the center there and a really fun and free element I added was this dead stick from the outdoors. I actually took it right off of a tree and I added it for some additional height, but I also love the organic nature that I added. And I added just a small piece of it on the left side as well to make it kind of feel like it was throughout the entire mantle area. Now these houses here are from Anthro as well. I have quite a bit of decor because I actually worked with them on an Instagram this season. So I love their holiday decor as well. I think they do such a great job. So I'll link below absolutely everything you see throughout the video. If you are curious on anything, I'll kind of do it per suction as well. But I did layer in a few different bits from anthropology, including this gold star garland and also these little like finial pieces are really fun as well, that are glass, they light up. So I popped a few of those throughout and added a few bells to the end there. And that was really how I finished this off. These stockings are really cute. And I love just adding bows to absolutely anything this holiday season. I feel like bows are the number one trend for the holidays this year. Now I've been blabbing too much throughout this video.  Every single thing you see on this tree is actually from my shop, which is so exciting. This is like the Lone Fox shop tree this year. And I added a bunch of these scalloped wood garlands which are so cute, paper mache ornaments and really filled it out. I wanted it to kind of have this folky just very crafty DIY feel to it. And I love the way it looked. And to wrap up the tree, just adding a couple pieces of the red silk ribbon as bows. And that finishes up this holiday season's decor. I cannot wait to reveal it to you.


The title of our sermon this morning is The Penalty Which Was Due. This is part three in this text, Romans chapter one, verses 26 through chapter two, verse one. The penalty which was due. Welcome back again to our verse by verse journey through the book of Romans. Today we're coming to the end of chapter one. I feel like we're arriving at a sad milestone coming to the end of chapter one. We only have 15 chapters left in this book together. If you're visiting with us, it's a great time to jump in. We are just getting started. We're coming to the end of chapter one. And although we arrive at this morning at the end of chapter one, we do not arrive at the end of Paul's argument here. And the intent of this section of text, Paul pays absolutely no attention to the chapter division that's there. Wasn't there when he wrote it. And so Paul just keeps going. And what we're going to do is we're going to keep right ongoing with him. And we're going to spend all of chapter two and most of chapter three developing Paul's opening statement. We're just getting started with Paul's opening statement. So if you consider chapter one, verses 26 through chapter two, verse one, the court has been called to order. You and I together with all mankind, we sit in the seat of the accused. And as a good prosecuting attorney, Paul is carefully now very carefully specifying the charge and he's beginning to lay out an airtight case against us. We are all guilty before God. There is none righteous. No, not one. There is none who does good. No, not one. There are none who understand. There are none who seek after God. The evidence against us is stacked up to heaven.  One of the concepts now that Paul is developing in this section of text is reciprocity, reciprocity. God's retributive justice, the wrath and judgment that God pours out on man against sin is reciprocal. It's the concept of reciprocity. Paul refers to this reciprocity in verse 27 as the penalty of their error which was due. That's a frightening statement on multiple levels. In other words, the penalty, the wrath that God pours out on man against his sin, against man for his sin is a proportionate response. It's an appropriate response. It's a measured response. It's fitting, the Bible would say, when you consider what we've done. Think with me for a moment about that statement. When you consider our sin against God, when you consider the wickedness that dwells within your own heart, your sin against him, God's wrath poured out against you for your sin is a measured, proportionate, appropriate fitting response. Let's unpack the implications of that for a minute. Think with me. God considers the depth of our sin. He ponders, if you will, our rebellion against him. He ponders our rejection of his gracious revelation. He considers that we would rather worship ourselves than to glorify him as God and to be thankful. He concludes that a measured response, a proportionate response, an appropriate response to that rejection is divine wrath. The wrath of an omnipotent God poured out full strength into the cup of his indignation, the wrath of God's abandonment in this life, the wrath of hell into eternity, and God says that's a proportionate measured response. What does that then say about the evil of our sin? What does that communicate about the wretchedness of our own condition as sinners? The unholiness of sin is in direct proportion to the holiness of God. What does that communicate about the holiness of God? What does that communicate about the ungodliness, the unholiness of our sin? Low views of God will lead to low views of sin. I would submit to you the vast majority of professing Christians today have such a deplorably low view of God, such that they have then a very high view of themselves, a very low view of their own sin. We need this kind of teaching from the apostle Paul, don't we? When you're witnessing to people often, they absolutely don't see it. I can never serve a God.  Low view of God, low view of God's holiness, very high view of man, not considering whatsoever the wretchedness of man's sin against God. I want to submit to you that we don't see our sin as evil as God sees it. We don't see our sin as evil as God sees it. We don't see our sin as the evil that it must be to warrant such reciprocity. Jeremiah Burroughs, there are manifold manifestations of God's displeasure against sin, which when they are spoken of and opened up unto the people who do not understand the dreadful evil that is in sin, they stand and wonder at it. And they think that these things are hard and severe things, reminded of the words of the apostle Paul quoting Isaiah when he says, marvel and perish, marvel and perish. Behold, God says, I will work a work in your days that you wouldn't believe it if I told it to you, speaking of God's wrath, speaking of God's judgment against sin. Burroughs goes on to say, when ministers reveal the threatenings of God against sin, these people say, God forbid, we hope God is more merciful than this. And all because they do not apprehend what dreadful evil there is in sin. The soul that apprehends and believes these particulars, which have been opened unto you, cannot but justify God when they hear the revelation and the manifestation of the wrath of God against sin. In other words, the more that we understand our own sin, the more that we understand the sinful condition of our own heart, the more that we understand our rebellion against him, the more that we will justify God as righteous in his judgments, do you see? The more that we'll see God as a holy and righteous and just in his judgments. And of course, all of that is the intention of Paul in this text. One of the intentions, one of Paul's purposes here in this text is to vindicate the justice of God in pouring out his wrath upon sinners by opening unto them the wickedness of their rebellion against him. Paul wants us to sit here this morning, you and I, to look at this text and to lay our hearts before open, wide open, bare before ourselves, before our own sight as it is in the sight of God. He wants us to see ourselves more the way that God sees us. And so he labors in this text, building his case. Most of chapter one, all of chapter two, most of chapter three, just to get to the point of saying your mouth is stopped, you have no case, you are inexcusable, you are guilty before God. Do you see? Vindicates God for his justice. What's the purpose of all that? So that you and I seeing the truth, understanding the truth would turn from our sin and would trust alone in Jesus Christ, who alone can save us from the wrath of God. Do you see? It's for us to be saved. Every bit of this is dripping with grace, do you see? Because it should point us to Jesus Christ. The one who has given us life and breath, the one in whom we live and move and have our being has graciously and sufficiently revealed himself to us. And what has man done with that gracious revelation? Man has rejected it. Rejected it out of hand. They're not just rejecting the revelation, they're rejecting the God who has revealed himself to us in it, do you see? Verse 18, the wrath of God has revealed from heaven against all this ungodliness, against this unrighteousness of men who suppress the truth and his unrighteousness. Because reciprocity, do you see? Reciprocity, because what may be known of God is manifest in them because God has shown it to them, written on our hearts, so to speak. Since the creation of the world, his invisible attributes clearly seen, clearly being understood by the things that are made such that we, brothers and sisters, we, men, women, boys, girls, we are without excuse. Verse 21, we know God and yet refuse to glorify him as God. We're not even thankful. Verse 23, we exchange God as the object of our worship for a man-made image. We plunge ourselves into idolatry. Therefore, reciprocity, reciprocity, God gave them up to uncleanness. Verse 24, they exchange the truth of God for a lie. Therefore, a reciprocal response, God gave them up to vile passions. Verse 26, they don't even want God in their thoughts.  Therefore, reciprocity. God gave them up to a debased mind. Verse 28, listen to Romans chapter two, verse five, reciprocity in accordance with your hardness and your impenitent heart. In other words, it's a proportionate response. It's a fitting response to your hardness, your impenitent heart that you are treasuring up for yourself wrath in the day of wrath and revelation of the righteous judgment of God. The righteous judgment of God. Who will reciprocity? Honor to each one according to his deeds, eternal life to those who buy patient continuance in doing good, seek for glory, honor and immortality, but to those who are self-seeking and do not obey the truth, but obey unrighteousness, indignation and wrath, tribulation and anguish on every soul of man who does evil of the Jew first and also of the Greek. But glory, honor, and peace to everyone who works what is good to the Jew first and also to the Greek. There is no partiality with God. God is just when he speaks. God is righteous when he judges. God is holy in his wrath. God is right to judge sinful man, is he not? We have a tendency to think that God sees our sin the way that we see our sin. We tend to justify our sin. We tend to minimize our sin. We excuse our sin. We ignore our sin. We forget our sin. We take pleasure in our sin, but how evil, how wicked must our sin be to consider as reciprocal the suffering and death of the sinless Son of God to quench the wrath of God against it? What does that communicate about the wretchedness of our sin? See who think of sin but lightly, nor suppose the evil great? Here may view its nature rightly. Here its guilt may estimate. Mark the sacrifice appointed. See who bears the awful load, tis the word, the Lord's anointed Son of Man and Son of God. You can look at the cross and see the wretchedness of our sin. At the same time you look at the cross to see the wretchedness of our sin, you see the glory of His grace, the glory of His love, His mercy, His condescension. These words are dripping with grace. Chapter 1, chapter 2, chapter 3, the world looks at this and they say, this is crazy. This is absurd.  Why would we ever come to church to hear this? You talk to people all over the place. They don't want to hear Hellfire brimstone preaching, but you don't want to hear biblical preaching. This is what the Bible tells us. And we need this. It's like, come on, Paul, give it to me. I've got to get this. Let me have it. Give it to me straight. Let me view my nature rightly that I might flee for salvation to Jesus Christ, the refuge of the cross. We need this truth and we need Paul to take his time. We're not in any rush, are we? We need Paul to take his time. Referring to your outline, we've considered point one in this text and the tragic refrain that expresses the retributive justice of God's wrath revealed from heaven against all ungodliness and unrighteousness of men. Three times, God gave them up to uncleanness. God gave them up to vile passions. God gave them over to a debased mind. We've considered point two on your outline. The shameless perversion to which they have been delivered. Men reject God, plunging themselves into idolatry, and then God gives them over. He delivers them up, abandons them to the vile and unnatural passions of their wicked heart. Ultimately, as we've seen in the text, manifesting itself in the degrading, desecrating abomination of homosexuality. Paul refers to homosexuality as the penalty of their error, which was due. It's reciprocity, isn't it, reciprocity? Consider with me now point three then on your outline, a corrupt compulsion. A corrupt compulsion. We see that beginning in verse 28. And even as they did not like to retain God in their knowledge, God gave them over, there's the language again, gave them over to a debased mind to do those things which are not fitting. In these verses, Paul once again continues to refer to the wrath of God's abandonment. This is the third time in the pattern. God delivers men up to the dominion of their sin, and they are now ruled by unnatural lusts. They've come under the dominion of vile passions, committing that which is shameful. The dominion of their heart leads to a dominion of their wretched conduct. Being already referred to the abomination of homosexuality as the most blatant evidence of that, of man's degeneracy, Paul now expands his evidence in this text to include an absolute deplorable list of all manner of vile vices, right? Now at the outset, I want you to see a very important connection in the text. It's not clear in our English translation, but it is there in the Greek and I want you to see it. Follow along with me. Paul says, verse 28, they did not like to retain God in their knowledge, okay? They did not, the word means approve. They did not approve. They did not find it fitting to retain God in their knowledge. The word is from dachymodzo, meaning to approve of something as good or to approve of something as fitting, to approve of it as appropriate, okay, dachymodzo. In other words, they found it unfitting to have God in their knowledge. And so God gave them over to an adachymos mind, an unfit mind, see what Paul is saying in the connection in the text, the word play that's going on here. They did not find it fitting to retain God in their knowledge, so God gave them over to an unfit mind. What do you see in that? Reciprocity. What do you see? Reciprocity. God's judgment, God's wrath is not arbitrary. It's not wild, given over to passions like the lusts or the anger of man. God gives sinful man exactly what sinful man deserves. He pours out a measured, a muted response. It doesn't mean that it's a light response. It is a hard response. It is a severe response and it is exactly what man deserves for his sin. Reciprocity is communicated really clearly, verse 28, in that little Greek word translated even as, even as, even as they did this, God as a fitting response did that, right? It's proportionate. It's appropriate. Dr. Murray says there's a correspondence between the sin committed and the recompense made it out. Arminians, most of professing evangelicalism, most professing Christians who aren't taught well tend to think of sin as this nebulous black blob, God's wrath as this sort of unreasonable response to all that, and salvation as being this thing that just makes heaven possible or potential. It really all depends on man accepting Jesus into his heart or God just simply forgiving because that's what God is supposed to do and somehow we just all make it to heaven. No. God is just, justice will be done. Not worldly justice like we see all over the place today. Not worldly conceptions of justice like people are propagating in our country today. God is perfect in his justice. Justice will be served. Every sin paid for, if they're not paid for by Jesus Christ, who do you think pays for them? You do in justice in eternity pray to the Lord Jesus Christ that he would save you from your sin, right? And bear that penalty for you. He's the only one who can. God will meet out a proportionate response to sin and he considers hell a proportionate response. You see? Even as they disapproved of having God in their knowledge, that which is understood or recognized by the mind, right? Knowledge is that which is understood and comprehended, recognized by the mind.  The faculty through which they attain knowledge then is disapproved. These are the same words used by Paul in 2 Corinthians, chapter 13, verse 5, where Paul says test yourselves. In other words, approve yourselves, dachimadzo. Examine yourselves as to whether you are in the faith. Do you not know yourselves that Jesus Christ is in you unless in fact you are a dachimas, right? Dachimadzo yourself so that you know you're not a dachimas, right? Test yourselves, approve yourselves in that sense, unless in fact you are disqualified. What is Paul saying here? These that reject the revelation of God, plunge themselves into idolatry and who are now objects of God's wrath, they did not love, they did not cherish, they did not value or rejoice in the knowledge of God. God has graciously condescended to reveal himself to us, they could care less.   The judgment of God poured out against the very seat of our thoughts and the rejecting mind becomes the rejected mind. More than that here and I want you to see this, more than that, the knowledge referred to you here in verse 28 is that knowledge which necessitates a response. We're not merely to know about God, right? We're to know God and when you know God, that knowledge of God produces fruit. It provokes, compels, constrains man to worship, constrains, compels us to adoration, to praise, to service, to devotion, to affection, to love, to rejoicing, to hope, to encouragement, right? That knowledge of God produces in us fruit. The knowledge of God in verse 21, if you look there, should have compelled men to glorify him as God and should have compelled them to be thankful. But they did not do that which is fitting given their knowledge of God. And so God delivers men over to do that which is not fitting, do you see? Reciprocity, this is all retributive justice, it's establishing reciprocity. We get what we deserve. That's justice, justice. What else does this communicate? What else? What we think is directly connected to how we act. What we know of God directly impacts how we live for him. How we live in his presence, quorum deo. How we live before the sight of God, how we live under his gaze. It's all impacted by our knowledge of him, what we know of him, what we think of him. All of life is theology in that sense, do you see? And I would love like if I could lovingly take you by the lapels and like shake into you a love of theology, a love of how God has revealed himself in his word, good books that get you into the Bible, reading the Bible and seeing out of the Bible wonders of what God has communicated to us. A love for theology that love for God will produce fruit in your life, will change the way that you live, change the way that you think, change desires and affections, change the way that you process things, change imaginations, change hopes and it changes you. It's not just information, it's information for transformation, do you see? We need to love theology and listen brothers and sisters, it's good to listen to podcasts, it's good to listen to sermons, that's a great way to soak in information, but we've got to be readers of God's word. We've got to be studyers of God's word. We've got to get ourselves into good theology. Not just the practical applications of that, right? Of course, Lord, I want to be a more faithful husband, a more faithful father. I want to raise my children in the way that honors the Lord. We want to do those things. We want to practically obey the Lord in all those things, but I tell you, my knowledge of him is going to make me a better husband. It's going to make me a better father. It's going to make me a better, right? It transforms who we are, how we think, how we live. What we think is directly connected to how we act. Verse 28, these godless men found it unfitting to know him, so God gives them over to an unfit mind to then do those things which are not fitting. Do you see the dots connected there? They found it unfitting to know him. He gives them over to an unfit mind to do those things which are not fitting, not proper to do those things which ought not to be done. Paul then follows that truth with a horrific list of desecrating sins that characterize those with a depraved or reprobate mind. The mind is the seat of thought, thought is the seat of action.  They are whispers, backbiter, haters of God, violent, proud, boasters, inventors of evil things, disobedient to parents, undiscerning, untrustworthy, unloving, unforgiving, unmerciful who knowing the righteous judgment of God and those who practice such things are deserving of death not only do the same but also approve of those who practice them. They're not all of those things all at once, mind you, right? This is a representative list, not a comprehensive or exhaustive list, and these who are given over in this way do things like these, okay? The Lord makes the same connection regarding Israel. Turn back, let's look at an example of this in Jeremiah, Jeremiah chapter 4. Jeremiah chapter 4, look there at verse 22, the Lord makes the same connection regarding Israel, we see examples of this all over the Bible, rejecting the knowledge of God leads to a debased mind and a debased mind leads to depraved action. What we think, how we think is directly connected to how we act, right? A debased mind leads to depraved action. Jeremiah chapter 4, verse 22, now the Lord here is speaking of his own people, his own people, these are not the gentile nations, the Lord is speaking to Judah. To Judah, he says, my people are foolish. They have not known me, that's interesting that he says that, these are the people of God, and God is saying to them, you don't know me, you don't know me. They are silly children, they have no understanding, therefore, reciprocity, they are wise to do evil, but to do good, they have no knowledge, right? Again, true knowledge of God is connected to how we live our lives, of course the nation of Israel had knowledge of who God is, right? They certainly knew about him, they had been raised learning about him, however, in the same way today, you can find people all over churches today, at this very moment, all over churches, all over the world who know God, so to speak, all over seminaries, you will find people who know about God, up and down your street, know all kinds of things about God, and yet have no knowledge of him, and yet do not know him, right? How many lexicons, how many commentaries written by godless unbelievers, Trinity denying, Deity of Jesus Christ denying, Bible denying liberals, all kinds of knowledge about the Bible, and yet are entirely devoid of any knowledge of God, and bound for hell, absolutely astonishing. The Lord says, my people, my people are foolish, they have not known me. True knowledge of God is a knowledge that is applied, right? Knowledge applied is wisdom, right? The knowledge of God that is an applied knowledge makes one wise. These people, God says, are fools, they don't truly know him. If they did truly know him, they would be wise, they're not wise. My people, he says, are foolish. He says they are silly children, meaning they lack judgment. They have no understanding, meaning they are incapable of comprehending or discerning the truth, and what is the fruit of this sad condition, not knowing God? What is the fruit of it? They show great skill and aptitude for evil. Great skill and aptitude, they are wise to do evil, but to do good they have no knowledge. They don't know how to conduct themselves in righteousness, they have no idea how to do it. I want you to listen to this by way of contrast, listen to this. The highest science, the loftiest speculation, the mightiest philosophy which can ever engage the attention of a child of God is the name, nature, person, work, the doings and the existence of the great God whom he calls his father. There is something exceedingly improving to the mind in a contemplation of the divinity. It is a subject so vast that all our thoughts are lost in its immensity, so deep that our pride is drowned in its infinity. Other subjects we can compass and grapple with, and then we feel assigned a kind of self-content and go our way with the thought, behold I am wise, but when we come to this master science finding that our plumb line cannot sound its depth and that our eagle eye cannot see its height, we turn away with the thought that vain man would say he is wise, but he's like a wild ass's cult with a solemn exclamation, I am of, but yesterday and know nothing, but no subject will tend more to humble the mind than thoughts of God. And the same subject that humbles the mind is the subject that expands it. Charles Spurgeon wrote that when he was 20 years old, 20 years old, far from silly or foolish I'd say, right? Jeremiah chapter nine verse 23, let not the wise man glory in his wisdom, let not the mighty man glory in his might, nor let the rich man glory in his riches, but let him who glories glory in this, that he understands and knows me, that I am the Lord exercising loving kindness, judgment and righteousness in the earth for in these I delight says the Lord. Well that's gonna have an impact on your life, isn't it? Impact on how you live, how you love, it's gonna have an impact on your worship, it's gonna have an impact on your devotion to serve him, it's gonna have an impact on your evangelism, it's gonna reach into every square inch of who you are, you have that kind of knowledge of God. I knew a young man years ago who loved to talk theology, loved to talk theology, would debate theology, talk about theology, every time we got together we're talking about theology, he would study, read, he would memorize scripture, evangelize, he was serving and he was a reprobate, did not know the Lord, not long after given over to a debased mind to do those things which are not fitting, plunged himself headlong into sin. A full blown apostate went headfirst deeply into drugs, crime, left his wife, left his children, plunged himself into sin and rebellion and recently died. Died as an apostate, died in his sin, died apart from the Lord and yet knew so much about him, that's shameful, right, shameful and a fearful thing. We need to pursue knowledge of God through faith in the Lord Jesus Christ who died and gave himself for us. It's the only way to go about that, reading your Bible, praying to God, God I want to know you, I want to love you more the way that I should. I want to forsake my sin, I want to serve you as I should, I want to have a heart for you, I want to be a man after God's own heart like David, right, Lord change me, transform me through your word, help me Lord to understand, help me to follow you as I should and departing from sin, forsaking sin to trust in him, independence upon his spirit, not on our own strength, not for our own pride, not for our own shameful aspirations, so to know him. Jesus Christ says, doesn't he, that eternal life is to know him and know the one whom he has sent. Truly knowing God will not only produce a love for theology, truly knowing God will produce a love for the one who is the subject of theology, a love for him, his person, his work, that love will constrain the one who believes in him to worship and to praise and to devotion and to zeal. It'll constrain or compel that one to heart holiness, to live a holy life. It will lead to comfort and encouragement and hope and joy and peace and faith. It will lead to a love for theology that leads to a greater knowledge of him for who he is, a greater maturity in the faith. True knowledge of God produces good fruit. Daniel 11 verse 32, the people who know their God shall be strong and carry out great exploits. I like that. Spurgeon said, there was something exceedingly improving to the mind in the knowledge of God. The opposite is also true. The opposite is also true. There's something exceedingly destructive to the soul in rejecting that knowledge of God. Back in Romans chapter 1 verse 29 then, even as they did not like to retain God in their knowledge, God gave them over to a debased mind to do those things which are not fitting, which are not proper.  Again, this is not an exhaustive list. It is merely a representative list and it is a tragic list. Verse 29, the extent or the depth of man's depravity, the scope of man's depravity is expressed by the description given at the beginning of verse 29, being filled, being filled. All unrighteousness there is generic and it is comprehensive, implying that the wicked list that follows is a representative list of what all unrighteousness they are involved in. It's included in all unrighteousness.  They are filled with it. Do you see how thoroughly, how completely they have come under the dominion of a demaced mind? Their cup now runneth over with filth. They're choking on it. Now some of you may know exactly what Paul is referring to there with a battle or struggle that you've had with sin or a particular sin. That you sometimes are just overrun by it. You feel like you're drowning in it and choking on it. That's what Paul is talking about here. Being filled with all unrighteousness of that thing, whatever it is from this list or from other things. We're not going to go into each of these words individually. I think the words are self-evident and self-explanatory, easy to define, suffice it to say, this is how the Spirit of God describes those, the actions of those who are abandoned by God. Do you see? The Spirit of God has the ultimate author of this list. So consider with me now the last point, point three on your notes, point three on your notes, and the corrupt compulsion then, the corrupt compulsion. The end of this list, verse 32, Paul concludes by saying that knowing the righteous judgment of God then, that those who practice such things are deserving of death, not only do they do the same, but also they approve of those who practice them. Now think with me, they know the righteous judgment of God, just like they know God in verse 21. Just like they know God in verse 21, refusing to glorify him as God, refusing to be thankful, they know here the righteous judgment of God. What is the righteous judgment of God? That those who practice these things are deserving of death. Now the Greek word translated righteous judgment there refers to commandment or ordinance. The word is literally ordinance or commandment. This is the commandment that they know. This is the commandment. Those who practice such things are deserving of death. That's what God commands. That is in accord with his righteous character. That's in accord with his holiness. That's in accord with his law. That is in accord with perfect justice. Those who do such things are deserving of death. And again, this is a representative list, not an exhaustive list. They know this. They know the commandment. In other words, this is a witness from conscience against them. They know these things that they are deserving of death, those who practice them. And so in other words here, they don't act in ignorance. They act in full knowledge under the watchful eye of an omnipresent omniscient God. They act in broad daylight as it were. They don't act in ignorance.  And yet they both do those things and approve of others who do them. When the Bible speaks here of deserving death, those who deserve such things or do such things or deserving of death, it's not merely speaking of physical death. You understand physical death is a shadow. It's a type. The fact that every person dies is an indication that we deserve death for our sin. And the death that the Bible is referring to as just desserts for our sin that will come to us in that day is an eternal death. The Bible calls it the second death, an eternal death in torment.  And it's amazing to me how few, you know, the only time I've ever been to a funeral that actually preaches the truth about that was one that I preached or one of the brothers here preached. It's the only time, the only time, I've been to a lot of funerals, but you're at a funeral at the death of someone. That is an object lesson in reality, folks, an object lesson in reality. And you stand there over the grave of someone who has passed through that veil, and then you have some liar in the pulpit, he's in a better place, you know. It's despicable. It's despicable. This is a terrifying truth. We face eternal death. Those that do such things are deserving of that judgment. They don't do it in ignorance. This is not something they fall into in a moment of weakness. They don't fall into it in a fit of passion. They're not overtaken by some temptation unaware. No, as Jeremiah would say, they're wise to do evil, wise to do evil. They're calculated in their evil. Whatever that evil is from this list or among other things, whatever it is, they're calculated in their evil. They determine to make a practice of things like these. Well, how do we know that? They determine to do it. I don't want righteousness, they would say to themselves, I don't want to follow the Lord. This is what I want.  And how is it that it's calculated in a very calculated way? They approve of those who do the same thing. They look around, they look at themselves, what they're doing. They determine to do that. And they look around and they approve of everybody else who's acting in the same way. As if there's some twisted, sick strength they sense in numbers, right? We're all, and don't you hear that when you witness, right? When you talk to people, you witness to people, someone will say, we're all sinners. We're all sinners. If that's the truth, then we're all going to hell. Paul is describing here an even deeper and a far more calloused degree of depravity, a depravity to which the debased mind leads. This is the low point I would submit to you. It's not the homosexuality, verse 27, verse 26. This is the low point in their degeneracy. This is the low point in their desecration. This is the basement in the rusted out tin shack that is man's lost condition. Do you see? And knowing the righteous judgment of God, knowing it that they deserve to die, they are compelled against all reason, compelled against all knowledge to do that, which will lead to their own destruction. It is certainly, is it not, a corrupt compulsion. This is a testimony to man's depravity, a testimony to our sinful will that apart from a miraculous work of God in the heart of man, to cause him to be born again, to renew his nature, apart from that work through the person in work of Jesus Christ, our compulsion is to rage against all wise counsel from God's word and do that thing, which will lead to our own destruction, compelled by our flesh to do it. Absolutely a staggering testimony of the deplorable wickedness of man's heart. Unlike those who see the wrath of God poured out against sin and respond with a hatred for it. Unlike those who consider the horror that sin must be in the eyes of God to warrant such a penalty and flee to Christ to turn from it.  That knowledge doesn't cultivate any hatred for sin. That knowledge doesn't cultivate any desire to repent from sin.  Paul is referring to the indisputable nature of their guilt before God. Paul is making an airtight case, an airtight case against the condition of man. It inevitably and inexorably leads to point four on your notes then and a righteous judgment, a righteous judgment.  This statement of Paul, he brings his thought, this first section of his thought, to a close with an inclusio. It's literary bookends, so to speak, bookends on either end of Paul's thought here, encompassing a predominant thought. Here's that thought. These are those who rejected the revelation of God given in creation, verse 18, even though it was manifest in them the law of God written upon their heart, even though God has shown it to them, what may be known of God clearly seen, these have suppressed that truth of God in their unrighteousness, and Paul concludes that they are without excuse, verse 21. That rejection of God's gracious revelation results in a devastating series of consequences. One, men are lost in idolatry, creating a God after their own image, men exchange the truth of God for the lie. Second, they are given up by God, abandoned to vile lust and uncleanness. Third, they degrade to themselves with all manner of abhorrent and despicable sin, such that now Paul concludes on the basis of their actions, chapter two, verse one, they are without excuse. In the beginning, rejecting God's revelation of himself in creation, they are without excuse. Paul concludes his thought with the same judgment. Therefore, you are inexcusable, oh man, do you see how Paul wraps up his point, right? Rejecting the revelation that God has given will lead to sin, will lead to rebellion resulting in judgment and condemnation.  Who is the you of verse one? Who is the you of verse one? Who is Paul referring to when he says, you are inexcusable, oh man? Have you noticed the transition here? Chapter two, verse one, he goes from speaking of mankind, man, we've even done that through these sermons, haven't we? Talking about mankind in general, using mankind in the third person, they, them, right? Paul makes the transition now in chapter two, verse one, to the second person, you. Makes a transition from the third person to the second person, you. You, you are inexcusable, oh man. Speaking of you and I specifically, it's as if Paul was speaking to you from this text in relation to what he's just concluded. In other words, have you been listening to this text and judging those who practice such things? Oh, the wickedness of those deplorable people, the vile heart of those poor lost souls, my how they deserve hell.  How could someone be capable of doing something like that? I sure hope somebody's listening to this sermon. He needs to be hearing this because, you know, when Paul says, you are guilty, I am guilty of the same transgression against God. Therefore, you. So we come to the end of this, this section of text in which Paul has laid out an airtight case for the universal guilt of mankind in their sin. And then Paul says, and you, me, we're a part of that. You're inexcusable. I'm inexcusable because we, judging another, we do the same thing.  Do you not remember the depth of your depravity? Do you not remember just how filthy when you, when you, when the Lord opened your eyes, right? And you look back on those things of which you are now ashamed. Does it not just grieve your heart, right? Just devastate you to think I was capable of that kind of hatred and that kind of anger and that kind of wickedness and that kind of sin, right?   Sick of living as an offense against him.   Sick of making a wreck of it. And so you turn from your sin. You trust Jesus Christ. And now, praise God. There's joy. There's peace. And there's hope. And there's encouragement. And there's love. And Jesus Christ has saved you and I am his and he is mine. I'll never sin again. Until the next minute. And now you're sinning against grace, right? Now you're sinning against knowledge. How wretched is our heart? Who will deliver me from this body of death? I thank God through Jesus Christ. Our Lord will be delivered. We practice the same things. And whatever you judge another, listen.  You do the same. We are without excuse. Do you acknowledge that this morning? If you're here this morning, you're listening to this, you're reading through this text and you say, no way, not me. This is stretching it. This is extreme. Do you not acknowledge it?  Do you not acknowledge that? Acknowledging that is the first step in repentance. It's the beginning of wisdom, right? Fear of God. Our only hope in this is answered in the remedy that Paul gives back in Romans chapter one, verse 16. Paul, the reason that Paul is not ashamed of the gospel of our Lord Jesus Christ is because the gospel is the remedy to this problem. It is God's provision for this wretchedness. It is our solution, God's solution to our sin problem. It's the way that we can be made right with God, reconciled to God. Paul says I'm not ashamed of the gospel of Jesus Christ because the gospel in it, the righteousness of God is revealed. That righteousness that we need to be reconciled to God, that righteousness that we need to, in order to stand before Him justifying, Paul says is given in the gospel and the gospel is the power of God to salvation. Apart from a gift of His righteousness, this list describes you and I. Although you and I have done these things because of the gift of His righteousness given through faith, this list doesn't mark us anymore. It's been taken off our ledger. Praise God. And by His grace, it was placed on His ledger and He bore the penalty that that deserves in His own body on the tree. My ledger in Him is clean. Praise God. Praise God. God, Paul tells us these things. Paul confronts us in our sin. He shoves it up our nose, so to speak in these chapters, so that we may turn and be saved so that we might flee to Jesus Christ. Having turned now, brothers and sisters, that we may see our condition rightly, boast not in ourselves, but in Him who is our righteousness. All praise, honor, and glory be to Him. Amen. Amen.


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record. All right, everyone. Welcome to this analysis seminar. It's a pleasure to have today Cristina from the University of Barcelona. She's going to be talking to us today about on the number of lower sets with the fixed card event. Cristina, glad to have you here. First of all, thank you very much for organizing this visit for everyone who was involved in this. Thank you very much. Okay, let's start with this definition of integer partition. So what of it? We have a natural number and then its representation as a sum of non-increasing positive integers. It's called an integer partition of n. So let's just consider some examples. Let's say mv and r18 and let us decompose in line over 7 plus 4 plus 3 plus 2 plus 1 and 9. So it is an integer partition of 18 and we can represent it also via the so-called non-diagram. How do we do this? Let me just draw this staircase actually. So the first layer will have seven steps here. The next one is 4, then you have 3, 2, 1, right. So this staircase is actually called the n diagram and this is a representation of this integer partition. It's easy to see that any integer partition can be thrown in this way and vice versa. If you have a n diagram, we can recover the partition. So the important thing is that everywhere here must go smaller and smaller. Okay, so we know a lot of things about this integer partitions. Most of all, we know the generating function but if you stop to think a little bit, you can find this very nature and you can understand why it's the generating function. And we know, so yeah, I didn't say that p2 of n is the number of integer partitions of n. I think I will say something about this too for the next slide. For now, just picture it. And we know the asymptotic. So by having the Ramanujan, we know that p2 of n is for the main part, this is the exponent square of n times some constant. Okay, now we go further and we have these play partitions. So basically it's the same, but now every term in the sum has two indices and you require this monotonicity in two directions. And we will see an example. So how should we understand this? Let's stop here a little bit. So we have mij and let us think that i will stay for the, say, the green axis cube and j will be the red one. And then the value of nij will be the blue axis. Okay. So what do we see? Okay, m11, what is this? So the very high column here is three cubes, right, the very beginning. So we have m11 is three. Then, okay, we fix the green axis and we go along the red one. So what do we have? The next thing is two, right? So m12 is two. Then you have m22, right? One and then m11, yes. What do we have? m13 is one and m14 is one. Okay, then we will go further on this green axis and we have two m11, right? And one and two one is two and two is one and two is three. The next one is just this two and one. So m31 is two and m32 is one and the last one is m41 is one. So we recover this plane partition from this figure here. So you can also know that if you just say, okay, let's be the green axis with the axis of values, then you still have a plane partition, right? But another one where there's still a plane partition, okay? And also that's what is important.  Well, basically, okay, if you fix one here, the first coordinate, you have a diagonal, right? So you have three and two. Oh, no, it's the wrong thing, right? Three and two and one, one. It is an integer partition, right? So every slice you get is an integer partition itself. And every next slice is smaller than the previous one and it's contained in it, right? So these are things that we have to understand before we do anything. And, okay, we know also the generating function here in this way. And we also know the estimate process. So here in the main part is exponent of m with our two thirds and some constant. So we don't have a D dimensional definition, but we already can make some conjectures about the rules on this value. Okay, let's see the definition. So D dimensional partition is basically when you have D indices for every term and you require this one ethnicity in each of these D directions here. Okay, I can ask me what about generating function? Okay, so far nothing. So there was a conjecture made in McMuffin. So you see here my minus k here. In the previous case, there was minus one. So his idea was that the right thing to put here is like binomial coefficients depending on this D. And so it was staying as a conductor for quite a long time. But then it was proved that it's the wrong thing, even for the case four. And however, there are some numerics that show that probably the order is the right one, but nothing more. So okay, we don't know anything about this about the generating function. And okay, now what's the lower set because my title was lower set. It is actually the visualization of this partition. So in one dimensional case, when the partition was one dimensional, we had this two dimensional picture here. The case of the in partitions, like two dimensional partitions, we had this, okay, yeah, we have this three dimensional picture here. Okay, so now this two in p12 m stands for the dimension of this young diagram. And then in the case of three, it's the dimension of this figure here. So D will stand for this, the dimension of visualization. Okay, but we didn't find this formula yet. Let us do this. So what's the lower sense? It's basically a set, you can define it. So if a sub x belongs to q, then any other point, which have, which had all the coordinates, so we could then better x must belong to ourselves. So it's lower sense. If you have this x belong to q, then a x prime, that is x i less than x i clear, then we must have x prime in the set also. Okay, so how do I understand this? So in this dimensional case, we saw this third case here. And the definition from the beginning was like, I have every layer here less and less and less, but I can define it differently. If I cube here, the set here, give me my set, then all the cubes in this rectangular, not being my set also. So then I define this lower sense, and I will work with them, and not with the partitions. Okay, this will be more convenient. So yeah, you can see that you can do both points, and you also can think of cubes like this. So I mean, it's difficult to understand like d for height values of d, but you can think like you have, you can draw this hyperplace, the coordinate hyperplates, and then every cube, every unit cube you have in your set, you're either touching another one in each direction, or touch the the wall that the hyperplates. Okay, so that's the constant. And yeah, so now d has the right to stay here because it's the dimension of this lower sense. So d dimensional lower sets correspond to d minus one dimensional partitions. Okay, we are interested in this number. Okay, why? Well, first of all, this, the, this structure of lower sets, surprisingly, it appears in a lot of physical settings, like crystal growth, and the microstates in regards to particle inputs, harmonics, oscillators, much special as to this space. But the thing is that this quantity usually appears in physical quantities. But okay, there are some things you can do, you can apply this in mathematics also, in approximation theory at least. So if you have, sometimes it's good to approximate some function by a trigonometric polynomials, and it may sense that the harmonics of your polynomials stay in the lower set. But just by some observation, you can, you can understand this, if you have some harmonic here, you should have all the harmonics, okay. So yeah, what do we know about this video? First of all, we know this theorem by Bracev-Rasevinarov, which was also proved in physical work. So if you fix the dimension when you leave, then you just see that your logarithm of p of n is up to a constant, and to the power one minus one over t. So that's what we saw in this two-dimensional case, three-dimensional case, right? So this is the right thing. However, they didn't care about the constant at all. So these four applications, which is not very good. And for the purposes, so Diet Riemann-Schadt and Siegfried-Consumeric, they provided some explicit values for these constants. So first of all, you can say that this thing here, you always can think of it as constant to the power d, to the power lower d. And if you assume that your n is largely not, say, constant to the power d, then this c1, you can think of it as a constant, actually, because when you write this thing down, you can understand that it's quite at least one, always. So then you are dealing with this lower constant here. So you can think of it as constant, because usually for applications, not always, but usually you're not n being large enough in terms of the k. But the problem with this c2, which is quite huge here, so the main results we'll be talking about today is this one. So once you have n being d to the power d squared, you can just obtain that your logarithm of pd is up to an absolute constant, n to the power 1 minus 1 over d. And if for some reasons you want to release this condition to d to the power d over d, then you can put here additional d squared. Okay, so I will tell some things about ideas on the book. And we will need to introduce a notion of available subset of a lower set. So what is it? Let's draw something here to the mention once again. Please, we will call a cube available if we can take it away without, like, disturbing the lower set structure. So if I take this cube away, the remaining set will be lower set. So this cube is available. This is not available, because if I take it away, I'll get a hole. I can't get a hole. So actually in this set we have just one, two, three cubes available. So this set I will call the maximum available subset of this cube. And this is the definition. And n of q will be the maximum available subset. This is n of q. Now it will be interesting in estimating this point. And the n of q can obtain is that if you have n large enough, like d to the power of d, and if you have a lower set, but cannot add, then you can see that the cardinal is always available subset is at most some absolute constant times n to the power of 1 minus 1 over d. Actually, this is what we would expect. Why? So imagine you have a very strange lower set, like very low here, very low here, I don't know, something here. Would you expect to have a big available subset? No, right? If you can take away these cubes and put them somehow here, and you'll get much more flexibility here, right? So the intuition is you should uniformize your lower set to get this available subset more, right? And then, well, which sets are the most uniform possible? Okay, this one, right? You can take this set, or this set, or this set, right? So actually, what these sets are? So I have x in my Q, if and only if the sum of its order is less or equal than some prescribed numbers, right? And if you stay thinking and calculating, you will see that these sets have exactly this, I mean, after some absolute constant, they have these available subsets. Okay, now we will not prove the lemma, but I suggest us to prove something weaker. Actually, not weaker because in the previous thing, we had this requirement that n must be large, but now we, without assuming anything on d and n, we will prove that it's less than d times n to the power of one minus one over d, okay? Okay, we will do this by induction on the damage. Once again, we'll start with this two-dimensional case, good one. Okay, I have some Q in two-dimensional case. I claim that we always have, if I have a point x2, x1, x2, then there must also be either this x1 minus x2, it will be large. This or x2 minus x1. Okay, if not, then I have a point here, which is this one. I must have all the points beneath, so I already have points from 0, 1, 2 and a 10, the first one and the second one the same, so I have at least this number of cubes in my set, right? It's not possible. So okay, I have either this or that. This means I can put my set into these two straps in reunion and let me call these parts, which belongs to this strap, the one and the other one will be Q2. So we can think that you can see that these two sets are lower sets, right? Because you just cut apart the weight and that's that's all. So you have Q belongs to the reunion of Q1 and Q2 and the same you can say about the available substance, right? If it's available here, it must be available here, right? So here it is. Here you go. And then the current out of this guy is less than the sum of current out of this here. But what can I say about this guy here? It has the first coordinate being either 1 or either 0 or 1 or 2 of these numbers. And it's not that I cannot take away two cubes which are in the same first coordinate. This is one of these, one of them is beneath and then they're ordered. So there will be a hole. So the maximum value of this guy is square root of L, right? And here the same. So I have that the current out of the hole line is x1 to square root of L. And that's what I wanted. Okay, now I need to proceed my induction somehow. And so let's begin to the picture then. So remember we noticed that we can slice our set so that it became this. Every slice is a lower set itself. And then each next slice is a subset of the groups. Okay, now what I do, I will choose an axis I want. I will slice my set and I will get slices of coordinate as L0, the N1 and so on until some say Nm. Okay. Okay, for each slice I can do something. For example, I can use the induction assumption to get it on case set. What I have, I have there was this, we're proving this. Now what I have with d-1 instead of d is this thing. But also remember that if I have the scale level, then I have table at 1. So this is a slice at here. And it belongs like it lies above this side. So if I take here away q, which also belong here, there will be a hole. Okay, so I can't do that. Also I must have at most this number of cubes. Okay, I have two estimates. This one from the induction and this one just from general reasons. Okay, now I can write the n of q is at most the sum of k from 0 to n, just for the sake of simplicity, let's take that and we'll start with the new. And here I have minimum for the first guy things and this difference. Now I will break this sum into two parts. So the first part will go from 0 to this number like m to the bar 1 over d minus 1. And I will use the first estimate here. And the next sum is just a scope. Okay, I will just use this and I will get m with this. What can I say about this? This is a slice of number m to the bar 1 over d and it's less than m0, it's less than m1 and so on. So at most this can be n divided by this number. So the least in the among all first n to the bar 1 over d slices. Okay, and this, what can I do? The sum of nk is at most n and this power is less than 1, so the maximum is when all they are the same. So I have m to the bar 1 over d things here and I have this way. I will not deal with the full function here because it's boring, but what we get is this thing and it's exactly what we want. So we have 1 here and d minus 1 here and we get what we want. Okay, so we wrote some simple version of the lemma and recorded in our original lemma we had m of d, but an absolute constant here. Okay, so if we denote this tm being the maximal cardinality of available subsets of all sets of cardinality m then when n is large enough we'll have this as and then what can we do? We can estimate the number of lower subset of a set. So if we know the number of cubes will take away from this set, the original set was with cardinality n and then we take away at most k cubes and the question is how many lower subsets we can get? So this is an estimate for this number which depends on this t of n which was the cardinality of available subsets. Then we plug in these estimates and we get something like this, but let me give any intuition how do we do this. So I have a lower set once again let me do the mention. It's lower subset just for the sake of whether it would be like this, a and u and it's subset. I want to come from a to b discarding some cubes here, step by step but not one by one. So how can I do this? I will mark all the available cubes so here I have this one, this one, this one, this one and I will see which ones I can take away and which ones must stay to form b at first. Okay so this one must stay, I will not touch it and this one also but these two I can remove. So I remove this one, this one I can choose from 8 primes. Once again I will see the available subsets, this cube, this cube and this two cubes. So these two cubes must stay forever so I will never touch them but they are participating in my available subsets always. So okay I will remove this one and this one and then once again I will see the available subset, this one and this one and I will discard them and finally I will discard this cube and I will come to b. So basically if I know the number of cubes I want to remove and if I know the cardinality of this maximal available subsets only in which step and I also remember that these two cubes they give some value, additional value into this cardinality of available subsets but they are never touched. So if I take all these things into account I can do something here and prove them. Okay basically then the idea is to split your initial lower set into some tricky way into different lower subsets to apply this estimator, this one and the level one after that but I will not enter into details here. So this is about this theory but now I'd like to say change a little bit the sub topic so we're going to talk a little bit about five dimensional lower sets. So remember here we had like m being constant to the power of d and sometimes like d to the power x squared okay forget about this. Now I don't know anything about the relations between n and d so what can I say from nothing I can say this. So this is the theorem by Cohen-Mirat and Navier they say that every in every case you have p of n that's the ripple down to the power 2 to the power dn and in independent estimate which is d to the power n minus 1 n minus 1 look for it. Okay so it turns out that the the second estimate can be refined it was done by that remark shall be taken with 0. So you can just discard this n minus 1 factorial which is that's right here and now just and you put this variable coefficient on the left hand side but now just for the sake of curiosity you have if you assume that you fix the current value of your set and you just write the dimension when you where you leave then d to the power n minus 1 is the right thing this n minus 1 factorial is just constant in this case so but this case is not reasonable but I mean if you assume this you have the right order. So now we will talk about the case when actually d is quite d so at least m divided by some power of m and we have some estimate in this case so first of all this is the very very unnatural case when d is greater than m cubed then you have super sharp estimate so the right hand side you have this variable coefficient from the previous slide and then you have you can estimate from above where the same thing doubled. Okay now there are three estimates the first case when d is much bigger than m squared than when d is of order m squared and then when d is much less than m squared but still with this property here basically you have this sharp order estimate for the lower and yes I would like to prove the first part because it works here. Okay how do you do this just a few minutes ago we were deconstructing these over sets and let us now construct it from the beginning so how do you do it you have n cubes you have to put them somehow so first of all I have no option I have to put the first cube at the origin okay but now you have still n minus 1 cubes and let me do the following I will choose any number j from 1 to pi minus 1 and then I will put j cubes along the x so come here here here okay how many ways I have to do this I have d axis and j cubes to put so it's the the binomial condition d minus 1 plus j choose minus 1 okay now I cannot put any cube more along the x so this is prohibited you know what does it mean any cube now must have at least two non-zero coordinates it means that it must lean on some two previous cubes right okay any two cubes they cannot generate more than one place to put another one and what does it mean I have just j cubes here and I have to choose two to put any one so at most I have this number of possibilities like j choose two okay the next step I already have j plus one cubes and the number of ways to put a new one is at most and so on and so forth I come to this thing so the maximum the maximum number of ways to construct any one was set is to sum this over all j and okay let's do now with this quantity here by a j and then if this is very big you can say that the every next a j is at least twice the previous one that means that I can estimate that the whole thing by twice there's the last guy so it makes sort of sense because if you have too many axes then it was putting cubes along them and not one into another right so that's the very rough thing but if you want to obtain other estimates you will have to do in a more delicate way so you have to create your steps differently so okay first of all you put this one cube at the origin but then I will put just the cubes which has the form zero zero one zero zero so I will choose the axes which will stay in my game and the other axes they will not participate anymore okay so for example I choose this one this one and the next step I will put all the cubes all the form zero zero one zero zero one zero zero or zero zero two zero zero okay so those are the cubes with the sum of coordinates equals two and each step will be just putting the cubes with the sum of x i being k and then you can see that each cube from this set must lean on the cubes with the sum of coordinates equals k minus one okay and this gives you I mean you can carefully estimate this and if you have like sk cubes so you don't have to read this if you have sk cubes on the k step you can estimate that the number of places to put new cubes that most this one so the number of cubes from the previous uh on the previous step squared over two and to choose then sk cubes you have this binomial equation the minus minus squared that's k and now you have to multiply all of them and so p of l will be the sum of these guys here and then the the problem becomes just purely analytical and then you have to analyze these quantities but yeah yeah for this all I wanted to say thank you very much this thing over there won't have questions or comments in the in sub step you mentioned that the cube is available in the sub in the subset uh can it be also available in the cube like in the in the original set uh like when you when you consider nk minus nk minus one yeah you use this fact that you cannot remove from the lower set like I was thinking there might be some cubes in the corner like in available cubes that that are available in both like uh no it's it's possible because they're orders so they are like for example in each of them mentioned this okay so if back here the cube which appears also here it cannot be available yeah that's true but if you just remove the one available cube from this you get the subs like subset yeah if you want to say that if you remove this one yes now that but the the definition of available cube is that you just remove this one and then it's available but if you remove if you have to remove another one in order to remove yours it's yeah what I'm talking about is not that one but the other available cubes uh for example the one on the lower corner this one yeah this is available in both cases right yes and how did you count that that available cubes in nk and nk minus one so I have here n1 yeah and zero it's just four right and three it's oh and one is three so I can just remove the cube which is here but not here and the number of such cubes is and the roman is in one here is just one but all these cubes have to stay because I have something leaning on them but this is just the cubes which appear here but not here and the number of them is that bit of difference okay thank you for the questions yeah you're talking about that part where she used an estimate of a meaning you know yeah one of the things that comes from and you comment a little bit on this generating function stuff for christine yeah beginning so what was the conjecture for the generating function of before so yeah so it has why it's actually doesn't work why it doesn't work I mean it says like what's the best since it doesn't work do you have any kind of new projectors about what the what generating function could be so the first case it was all right yeah it had product here from minus k and here i had minus one so in the second case it had minus k so the conjecture was so this is like two-dimensional things this is three-dimensional and here we have the power which is the binomial coefficient right k zero so here you have k one so the the conjecture was that you have k d minus minus two okay like this but it took a long time to understand that it's not the right thing so there are a lot of numerics and i mean no addition for me at least i'll be from there there's no conjecture just for the case four for the next case which is unknown i don't think so possible generating function yeah that's very interesting so there are people who just they do these numerics and it's not not so easy this gets complicated very fast right the numbers get big very fast yeah and there are tables i mean you can find some tables for quite huge numbers like in six dimensions but i mean that doesn't give you a clue how to think of it it's formal of rtm ramada channel it's very nice okay more questions well if not let's thank you christine again for the nice talk thank you christina thank you