Intercalates and discrepancy in random Latin squares

Abstract

An intercalate in a Latin square is a 2 × 2 Latin subsquare. Let bold-italicN be the number of intercalates in a uniformly random n × n Latin square. We prove that asymptotically almost surely N≥true(1−otrue(1true)true) n2/4, and that EN≤true(1+otrue(1true)true) n2/2 (therefore asymptotically almost surely N≤fn2 for any f→∞). This significantly improves the previous best lower and upper bounds. We also give an upper tail bound for the number of intercalates in 2 fixed rows of a random Latin square. In addition, we discuss a problem of Linial and Luria on low‐discrepancy Latin squares.

Almost all Steiner triple systems are almost resolvable

“…Combining the ideas in Section 7.2 and [31, Section 5.1.2], we have essentially proved the following theorem bounding the discrepancy of a random Steiner triple system, which may be of independent interest. We remark that an analogous theorem for Latin squares (with a stronger error term) was proved by Kwan and Sudakov [32]. See also the related conjectures in [33].…”

“…Another interesting question (also mentioned in [31]) is whether a random Steiner triple system typically contains a Steiner triple subsystem on fewer vertices. McKay and Wanless [34] proved that almost all Latin squares have many small Latin subsquares (see also [32]), but it was conjectured by Quackenbush [38] that most Steiner triple systems do not have proper subsystems. By comparison with a binomial random 3-graph, it seems likely that this conjecture is actually false, but it seems that substantial new ideas would be required to prove or disprove it.…”

Almost all Steiner triple systems have perfect matchings

“…• Another interesting question about random Steiner triple systems is whether they contain Steiner triple subsystems on fewer vertices. McKay and Wanless [36] proved that almost all Latin squares have many small Latin subsquares (see also [32]), but it was conjectured by Quackenbush [39] that most Steiner triple systems do not have proper subsystems. It seems unlikely that the methods in this paper will be able to prove or disprove this conjecture without substantial new ideas.…”

“…We note that in the case of regular graphs a much stronger phenomenon occurs: there is a sense in which a random ‐regular graph is ‘asymptotically the same’ as a random ‐regular graph combined with a random ‐regular graph (see [22, Section 9.5]). Another interesting question about random Steiner triple systems is whether they contain Steiner triple subsystems on fewer vertices. McKay and Wanless [36] proved that almost all Latin squares have many small Latin subsquares (see also [32]), but it was conjectured by Quackenbush [39] that most Steiner triple systems do not have proper subsystems. It seems unlikely that the methods in this paper will be able to prove or disprove this conjecture without substantial new ideas.…”

“…Glebov and Luria [19] gave a simpler entropy-based proof of the same fact and asked whether the counterpart of Theorem 1.1 holds: Do almost all Latin squares have essentially the maximum possible number of transversals? Although there do exist a number of techniques for studying random Latin squares (see, for example, [9,12,20,32,36]), none of them seem suitable to attack this question. In the time since the first version of this paper, Keevash [28] has generalised his methods to a number of different classes of designs, including Latin squares.…”