Almost all Steiner triple systems are almost resolvable

Ferber, Asaf; Kwan, Matthew

doi:10.1017/fms.2020.29

Cited by 21 publications

(22 citation statements)

References 36 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“… We believe the most interesting problem that seems approachable by our methods is to prove that almost all Steiner triple systems (and Latin squares) can be decomposed into disjoint perfect matchings (transversals). Since the first version of this paper, together with Ferber we [17] proved an approximate version of this fact: namely, almost all Steiner triple systems have

false(1 - o (1) false) n / 2

disjoint perfect matchings. As mentioned in the introduction, a Steiner triple system that can be decomposed into perfect matchings is called a Kirkman triple system, and even proving the existence of Kirkman triple systems was an important breakthrough.…”

Section: Discussionmentioning

confidence: 99%

Almost all Steiner triple systems have perfect matchings

Kwan

2020

Proc. London Math. Soc.

Self Cite

View full text Add to dashboard Cite

We show that for any n divisible by 3, almost all order-n Steiner triple systems have a perfect matching (also known as a parallel class or resolution class). In fact, we prove a general upper bound on the number of perfect matchings in a Steiner triple system and show that almost all Steiner triple systems essentially attain this maximum. We accomplish this via a general theorem comparing a uniformly random Steiner triple system to the outcome of the triangle removal process, which we hope will be useful for other problems. Our methods can also be adapted to other types of designs; for example, we sketch a proof of the theorem that almost all Latin squares have transversals.

show abstract

false(1 - o (1) false) n / 2

Section: Discussionmentioning

confidence: 99%

Almost all Steiner triple systems have perfect matchings

Kwan

2020

Proc. London Math. Soc.

Self Cite

View full text Add to dashboard Cite

show abstract

“…We remark that one can prove a similar coupling lemma for monotone decreasing properties (see [22,Lemma 2.6]), though this will not be necessary for us.…”

Section: A Coupling Lemmamentioning

confidence: 99%

“…• A Steiner triple system of order n is a 3-uniform hypergraph on a vertex set of size n, such that every pair of vertices is included in exactly one edge. These objects are natural "non-partite" analogues of Latin squares, and are even more difficult to study (to our knowledge, the only nontrivial results about random Steiner triple systems can be found in [2,22,40,51]). In the setting of Steiner triple systems, the 4-edge hypergraph we have been calling an intercalate is usually called a Pasch configuration.…”

Section: Introductionmentioning

confidence: 99%

Large deviations in random Latin squares

Kwan¹,

Sah²,

Sawhney³

2021

Preprint

Self Cite

View full text Add to dashboard Cite

In this note, we study large deviations of the number N of intercalates (2 × 2 combinatorial subsquares which are themselves Latin squares) in a random n × n Latin square. In particular, for constant δ > 0 we prove that Pr(N ≤ (1 − δ)n 2 /4) ≤ exp(−Ω(n 2 )) and Pr(N ≥ (1 + δ)n 2 /4) ≤ exp(−Ω(n 4/3 (log n) 2/3 )), both of which are sharp up to logarithmic factors in their exponents. As a consequence, we deduce that a typical order-n Latin square has (1 + o(1))n 2 /4 intercalates, matching a lower bound due to Kwan and Sudakov and resolving an old conjecture of McKay and Wanless. Kwan was supported by NSF grant DMS-1953990. Sah and Sawhney were supported by NSF Graduate Research Fellowship Program DGE-1745302.1 To see the analogy to permutation matrices, note that a Latin square can equivalently, and more symmetrically, be viewed as an n × n × n zero-one array such that every axis-aligned line sums to exactly 1.2 Jacobson and Matthews [30] and Pittenger [48] designed Markov chains that converge to the uniform distribution, but it is not known whether these Markov chains mix rapidly.

show abstract

“…In 4, we saw that the results of Luria [32] and Keevash [26] imply that the average Some evidence that this may not be straightforward to resolve is provided in [7], where Cavenagh and Wanless showed that, for almost all even n, there are at least n (1−o(1))n 2 Latin squares of order n without a transversal, let alone an orthogonal mate. However, Ferber and Kwan [16] study the analogous question in Steiner triple systems, and show that almost all Steiner triple systems are almost resolvable. In the context of Latin squares, they suggest that their methods would show that almost all Latin squares have (1 − o(1))n disjoint transversals.…”

Section: Orthogonal Matesmentioning

confidence: 99%

Enumerating extensions of mutually orthogonal Latin squares

Boyadzhiyska

Das

Szabó

2020

Des. Codes Cryptogr.

View full text Add to dashboard Cite

Two $$n \times n$$ n × n Latin squares $$L_1, L_2$$ L 1 , L 2 are said to be orthogonal if, for every ordered pair (x, y) of symbols, there are coordinates (i, j) such that $$L_1(i,j) = x$$ L 1 ( i , j ) = x and $$L_2(i,j) = y$$ L 2 ( i , j ) = y . A k-MOLS is a sequence of k pairwise-orthogonal Latin squares, and the existence and enumeration of these objects has attracted a great deal of attention. Recent work of Keevash and Luria provides, for all fixed k, log-asymptotically tight bounds on the number of k-MOLS. To study the situation when k grows with n, we bound the number of ways a k-MOLS can be extended to a $$(k+1)$$ ( k + 1 ) -MOLS. These bounds are again tight for constant k, and allow us to deduce upper bounds on the total number of k-MOLS for all k. These bounds are close to tight even for k linear in n, and readily generalise to the broader class of gerechte designs, which include Sudoku squares.

show abstract

Almost all Steiner triple systems are almost resolvable

Abstract: We show that for any n divisible by 3, almost all order-n Steiner triple systems admit a decomposition of almost all their triples into disjoint perfect matchings (that is, almost all Steiner triple systems are almost resolvable).

Cited by 21 publications

References 36 publications

Almost all Steiner triple systems have perfect matchings

Almost all Steiner triple systems have perfect matchings

Large deviations in random Latin squares

Enumerating extensions of mutually orthogonal Latin squares

Contact Info

Product

Resources

About