Large sets of t-designs through partitionable sets: A survey
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paper_title: On the maximum number of disjoint triple systems

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Let D(λ;v) denote the maximum number of mutually disjoint S(λ;2,3,v). We prove that D(2;v)=v−2/2 for all v ≡ 0 or 4 (mod 6), v ≡ 0 and that D(6;v)=v−2/6 for all v ≡ 2 (mod 6).
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paper_title: Some balanced complete block designs

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Letq=6t±1, ν=2q+2. The (ν/3) tri ples on ν marks may be partitioned intoq sets, each forming a BIBD of parameters (ν,3,2). Related results, some of them known, are also discussed briefly.
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paper_title: Covering all triples on n marks by disjoint Steiner systems

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Abstract Let q be a number all whose prime factors divide integers of the form 2 s − 1, s odd. If n = q + 2, the ( 3 n ) triples on n marks can be partitioned into q sets, each forming a Steiner triple system.
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paper_title: On large sets of disjoint steiner triple systems, IV

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Abstract To construct large sets of disjoint STS(3n) (i.e., LTS(3n)), we introduce a new kind of combinatorial designs. Let S be a set of n elements. If x ∈ S, we denote an n × n square array on S by Ax, if for every w ∈ S\{x} the following conditions are satisfied: Ax = [ayz(x)](y, z ∈ S), axx(x) = x, aww(x) ≠ w, axw(x) = axw(x) = x, and {awz(x) | z ∈ S} = {ayw(x) | y ∈ S} = S. Let j ∈ {1, 2&} , Aj = {ayz[j](y, z ∈ S) be a Latin square of order n based on S with n parallel transversals including the diagonal one. Two squares Ax and Ax′ on the same S are called disjoint, if ayz(x) ≠ ayz(x′) whenever y, z ∈ S\{x, x′}; two squares Ax and Aj on the same S are called disjoint, if ayz(x) ≠ ayz[j] whenever y, z ∈ S\ {x}; and two squares A1 and A2 on the same S are called disjoint, if ayz[1] ≠ ayz[2] whenever y ≠ez. It is a set of n + 2 pairwise disjoint squares Ax (x runs over S), A1 and A2 on S as mentioned above that is very useful to construct LTS(3n), and such a set we denote by LDS(n). The essence in the relation between LDS(n) and LTS(3n) is the following theorem which is established in the Section 2: Theorem . If there exist both an LDS(n) and an LTS(n + 2), then there exists an LTS(3n) also. The set of integers n for which LDS(n) exist is denoted by D. In the other parts of this paper, the following results are given: 1. (1) If n ∈ D, and q = 2α (α is an integer greater than 1), or q ∈ {;5, 7, 11, 19}, then qn ∈ D. 2. (2) If pα is a prime power, p > 2 and pα ∈ D, then 3pα ∈ D. 3. (3) If q is a prime power greater than 4 and 1 + n ∈ D, then 1 + qn ∈ D. 4. (4) If t is a nonnegative integer, then 7 + 12t ∈ D and 5 + 8t ∈ D.
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paper_title: Halving the Complete Design

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Abstract Let X be a finite set of cardinality v. We denote the set of all k -subsets of X by In this paper we consider the problem of partitioning into two parts of equal size, each of which is the block set of a 2- (v k, λ) design. We determine necessary and sufficient conditions on v for the existence of such a partition when k=3 or 4. We also construct partitions for higher values of fc and infinitely many values of v. The case where k=3 has been solved for all values of v by Dehon [5]. The case where v=2k has been solved for all values of k by Alltop [1], The remaining results are new. The technique used is similar to that used by Denniston in his construction of a 4-(12,5,4) design without repeated blocks. We also prove an interesting corollary to Baranyai's theorem giving necessary and sufficient conditions for the existence of a partition of into 1- (v, k ,λ) designs.
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paper_title: All block designs with b= v k /2 exist

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Abstract In this paper, we establish the correctness of a conjecture by Hartman on the existence of large sets of size two for t = 2. Also, we obtain some partial results for t > 2.
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paper_title: Extending t-designs

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Abstract Three extension theorems for t-designs are proved; two for t even, and one for t odd. Another theorem guaranteeing that certain t-designs be (t + 1)-designs is presented. The extension theorem for odd t is used to show that every group of odd order 2k + 1, k ≠ 2r − 1, acts as an automorphism group of a 2-(2k + 2, k + 1, λ) design consisting of exactly one half of the (k + 1)-settled, Although the question of the existence of a 6-(14, 7, 4) design is not settled, certain requisite properties of the 4-designs on 12 elements derived from such a design are established. All of these results depend heavily upon generalizations of block intersection number equations of N. S. Mendelsohn.
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paper_title: Parallelisms of complete designs

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Introduction 1. The existence theorem Appendix: the integrity theorem for network flows 2. The parallelogram property Appendix: the binary perfect code theorem Appendix: association schemes and metrically regular graphs 3. Steiner points and Veblen points Appendix: Steiner systems 4. Minimal edge-colourings of complete graphs Appendix: latin squares, SDRs and permanents 5. Biplanes and metric regularity Appendix: symmetric designs 6. Automorphism groups Appendix: multiply transitive groups 7. Resolutions and partition systems Bibliography Index.
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paper_title: Extending t-designs

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Abstract Three extension theorems for t-designs are proved; two for t even, and one for t odd. Another theorem guaranteeing that certain t-designs be (t + 1)-designs is presented. The extension theorem for odd t is used to show that every group of odd order 2k + 1, k ≠ 2r − 1, acts as an automorphism group of a 2-(2k + 2, k + 1, λ) design consisting of exactly one half of the (k + 1)-settled, Although the question of the existence of a 6-(14, 7, 4) design is not settled, certain requisite properties of the 4-designs on 12 elements derived from such a design are established. All of these results depend heavily upon generalizations of block intersection number equations of N. S. Mendelsohn.
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paper_title: Partitions of the 4-subsets of a 13-set into disjoint projective planes

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Abstract For 1⩽ t k v , let S ( t , k , v ) denote a Steiner system and let P k ( v ) be the set of all k -subsets of the set {1,2,…, v }. We partition P 4 (13) into 55 mutually disjoint S (2, 4, 13)'s (projective planes). This is the first known example of a complete partition of P k ( v ) into disjoint S ( t , k , v )'s for k ⩾4 and t ⩾2.
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paper_title: Parallelisms of complete designs

paper_content:

Introduction 1. The existence theorem Appendix: the integrity theorem for network flows 2. The parallelogram property Appendix: the binary perfect code theorem Appendix: association schemes and metrically regular graphs 3. Steiner points and Veblen points Appendix: Steiner systems 4. Minimal edge-colourings of complete graphs Appendix: latin squares, SDRs and permanents 5. Biplanes and metric regularity Appendix: symmetric designs 6. Automorphism groups Appendix: multiply transitive groups 7. Resolutions and partition systems Bibliography Index.
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paper_title: New Large Sets of t-Designs

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Abstact: We introduce generalizations of earlier direct methods for constructing large sets of t-designs. These are based on assembling systematically orbits of t-homogeneous permutation groups in their induced actions on k-subsets. By means of these techniques and the known recursive methods we construct an extensive number of new large sets, including new infinite families. In particular, a new series of LS[3](2(2 + m), 8·3m − 2, 16·3m − 3) is obtained. This also provides the smallest known ν for a t-(ν, k, λ) design when t ≥ 16. We present our results compactly for ν ≤ 61, in tables derived from Pascal's triangle modulo appropriate primes. © 2000 John Wiley & Sons, Inc. J Combin Designs 9: 40–59, 2001
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paper_title: Halving the Complete Design

paper_content:

Abstract Let X be a finite set of cardinality v. We denote the set of all k -subsets of X by In this paper we consider the problem of partitioning into two parts of equal size, each of which is the block set of a 2- (v k, λ) design. We determine necessary and sufficient conditions on v for the existence of such a partition when k=3 or 4. We also construct partitions for higher values of fc and infinitely many values of v. The case where k=3 has been solved for all values of v by Dehon [5]. The case where v=2k has been solved for all values of k by Alltop [1], The remaining results are new. The technique used is similar to that used by Denniston in his construction of a 4-(12,5,4) design without repeated blocks. We also prove an interesting corollary to Baranyai's theorem giving necessary and sufficient conditions for the existence of a partition of into 1- (v, k ,λ) designs.
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paper_title: Large sets of 3-designs from psl(2, q), with block sizes 4 and 5

paper_content:

We determine the distribution of 3−(q + 1,k,λ) designs, with k ϵ {4,5}, among the orbits of k-element subsets under the action of PSL(2,q), for q ϵ 3 (mod 4), on the projective line. As a consequence, we give necessary and sufficient conditions for the existence of a uniformly-PSL(2,q) large set of 3−(q + 1,k,λ) designs, with k ϵ {4,5} and q ≡ 3 (mod 4). © 1995 John Wiley & Sons, Inc.
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paper_title: A Few More Large Sets of t-Designs

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We construct several new large sets of t-designs that are invariant under Frobenius groups, and discuss their consequences. These large sets give rise to further new large sets by means of known recursive constructions including an infinite family of large sets of 3− (v, 4, λ) designs. c © 1998 John Wiley & Sons, Inc. J Combin Designs 6: 293–308, 1998
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paper_title: Locally trivial t-designs and t-designs without repeated blocks

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We simplify our construction [12] of non-trivial t-designs without repeated blocks for arbitrary t. We survey known results on partitions of the set of all (t + l)-subsets of a u-set into S(λ; t, t + I, μ) for the smallest λ allowed by the obvious necessary conditions. We also obtain some new results on this problem. In particular, we construct such partitions for t = 4 and k = 60 whenever ν = 60u + 4, u a positive integer with gcd(u, 60) = I or 2. Sixty is the smallest possible λ for such ν.
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paper_title: More on halving the complete designs

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Abstract A large set of disjoint S (λ; t , k , v ) designs, denoted by LS (λ; t , k , v ), is a partition of k -subsets of v -set into S (λ; t , k , v ) designs. In this paper, we develop some recursive methods to construct large sets ot t -designs. As a consequence, we show that a conjecture of Hartman on halving complete designs is true for t = 2 and 3 ⩽ k ⩽ 15.
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paper_title: More on the Existence of Large Sets of t-Designs of Prime Sizes

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In this paper, we employ the known recursive construction methods to obtain some new existence results for large sets of t-designs of prime sizes. We also present a new recursive construction which leads to more comprehensive theorems on large sets of sizes two and three. As an application, we show that for infinitely many values of block size, the trivial necessary conditions for the existence of large sets of 2-designs of size three are sufficient.
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paper_title: Extending t-designs

paper_content:

Abstract Three extension theorems for t-designs are proved; two for t even, and one for t odd. Another theorem guaranteeing that certain t-designs be (t + 1)-designs is presented. The extension theorem for odd t is used to show that every group of odd order 2k + 1, k ≠ 2r − 1, acts as an automorphism group of a 2-(2k + 2, k + 1, λ) design consisting of exactly one half of the (k + 1)-settled, Although the question of the existence of a 6-(14, 7, 4) design is not settled, certain requisite properties of the 4-designs on 12 elements derived from such a design are established. All of these results depend heavily upon generalizations of block intersection number equations of N. S. Mendelsohn.
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paper_title: Large sets of 3-designs from psl(2, q), with block sizes 4 and 5

paper_content:

We determine the distribution of 3−(q + 1,k,λ) designs, with k ϵ {4,5}, among the orbits of k-element subsets under the action of PSL(2,q), for q ϵ 3 (mod 4), on the projective line. As a consequence, we give necessary and sufficient conditions for the existence of a uniformly-PSL(2,q) large set of 3−(q + 1,k,λ) designs, with k ϵ {4,5} and q ≡ 3 (mod 4). © 1995 John Wiley & Sons, Inc.
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paper_title: All block designs with b= v k /2 exist

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Abstract In this paper, we establish the correctness of a conjecture by Hartman on the existence of large sets of size two for t = 2. Also, we obtain some partial results for t > 2.
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paper_title: More on the Existence of Large Sets of t-Designs of Prime Sizes

paper_content:

In this paper, we employ the known recursive construction methods to obtain some new existence results for large sets of t-designs of prime sizes. We also present a new recursive construction which leads to more comprehensive theorems on large sets of sizes two and three. As an application, we show that for infinitely many values of block size, the trivial necessary conditions for the existence of large sets of 2-designs of size three are sufficient.
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paper_title: Root cases of large sets of t-designs

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A large set of t-(v, k, λ) designs of size N, denoted by LS[N](t, k, v), is a partition of all k-subsets of a v-set into N disjoint t-(v, k, λ) designs, where N = (v-t/k-t)/λ. A set of trivial necessary conditions for the existence of an LS[N](t, k, v) is N|v-t/k-t for i = 0,.....,t. In this paper we extend some of the recursive methods for constructing large sets of t-designs of prime sizes. By utilizing these methods we show that for the construction of all possible large sets with the given N, t, and k, it suffices to construct a finite number of large sets which we call root cases. As a result, we show that the trivial necessary conditions for the existence of LS[3](2, k, v) are sufficient for k ≤ 80.
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paper_title: Halving the Complete Design

paper_content:

Abstract Let X be a finite set of cardinality v. We denote the set of all k -subsets of X by In this paper we consider the problem of partitioning into two parts of equal size, each of which is the block set of a 2- (v k, λ) design. We determine necessary and sufficient conditions on v for the existence of such a partition when k=3 or 4. We also construct partitions for higher values of fc and infinitely many values of v. The case where k=3 has been solved for all values of v by Dehon [5]. The case where v=2k has been solved for all values of k by Alltop [1], The remaining results are new. The technique used is similar to that used by Denniston in his construction of a 4-(12,5,4) design without repeated blocks. We also prove an interesting corollary to Baranyai's theorem giving necessary and sufficient conditions for the existence of a partition of into 1- (v, k ,λ) designs.
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paper_title: All block designs with b= v k /2 exist

paper_content:

Abstract In this paper, we establish the correctness of a conjecture by Hartman on the existence of large sets of size two for t = 2. Also, we obtain some partial results for t > 2.
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paper_title: More on the Existence of Large Sets of t-Designs of Prime Sizes

paper_content:

In this paper, we employ the known recursive construction methods to obtain some new existence results for large sets of t-designs of prime sizes. We also present a new recursive construction which leads to more comprehensive theorems on large sets of sizes two and three. As an application, we show that for infinitely many values of block size, the trivial necessary conditions for the existence of large sets of 2-designs of size three are sufficient.
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paper_title: Root cases of large sets of t-designs

paper_content:

A large set of t-(v, k, λ) designs of size N, denoted by LS[N](t, k, v), is a partition of all k-subsets of a v-set into N disjoint t-(v, k, λ) designs, where N = (v-t/k-t)/λ. A set of trivial necessary conditions for the existence of an LS[N](t, k, v) is N|v-t/k-t for i = 0,.....,t. In this paper we extend some of the recursive methods for constructing large sets of t-designs of prime sizes. By utilizing these methods we show that for the construction of all possible large sets with the given N, t, and k, it suffices to construct a finite number of large sets which we call root cases. As a result, we show that the trivial necessary conditions for the existence of LS[3](2, k, v) are sufficient for k ≤ 80.
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Title: Large sets of t-designs through partitionable sets: A survey

Section 1: Introduction
Description 1: Introduce the topic of large sets of t-designs, provide historical background, significance, and the scope of the paper.

Section 2: Definitions and Preliminaries
Description 2: Define key terms like t-design, block, simple design, and necessary conditions for existence. Introduce essential notation and provide preliminary examples.

Section 3: Review of the known large sets
Description 3: Give a brief account of known results on the existence of large sets of t-designs using various methods before focusing on partitionable sets.

Section 4: The necessary conditions
Description 4: Discuss the necessary conditions for the existence of LS[N](t, k, v) and provide alternative descriptions and examples.

Section 5: The approach of partitionable sets
Description 5: Explain the approach of (N, t)-partitionable sets and its significance. Introduce key lemmas and the method for constructing large sets using partitionable sets.

Section 6: General recursive constructions
Description 6: Present recursive constructions for large sets of any size using the approach of (N, t)-partitionable sets.

Section 7: Large sets of prime sizes
Description 7: Discuss recursive constructions specifically for large sets of prime sizes and highlight key theorems and their applications.

Section 8: Root cases of large sets of prime sizes
Description 8: Define root cases and show how specific large sets suffice for constructing larger sets, present important theorems and proofs.

Section 9: More results on large sets of sizes two and three
Description 9: Provide detailed results and constructions for large sets of sizes two and three. Discuss related theorems and their implications.

Section 10: Existence results
Description 10: Summarize the existence results obtained using partitionable sets, discuss significant conjectures, and present the best known results related to these conjectures.

Section 11: Open Problems
Description 11: Present and discuss open problems in the field, particularly focusing on determining root cases for large sets of various sizes and generalizing theorems for prime power sizes.