Discrepancy of high-dimensional permutations

Abstract

Abstract. Let L be an order-n Latin square. For X, Y, Z ⊆ {1, ..., n}, let L(X, Y, Z) be the number of triples i ∈ X, j ∈ Y, k ∈ Z such that L(i, j) = k. We conjecture that asymptotically almost every Latin square satisfiesholds asymptotically almost surely (this bound is obviously tight). We show that there exist Latin squares with ε(L) = O(n 2 ), and that ε(L) = O(n 2 log 2 n) for almost every order-n Latin square. On the other hand, we recall that ε(L) ≥ Ω(n 33/14 ) if L is the multiplication table of an order-n group. We also show the existence of Latin squares in which every empty cube has side length O((n log n) 1/2 ), which is tight up to the √ log n factor. Some of these results extend to higher dimensions. Many open problems remain.

Almost all Steiner triple systems have perfect matchings

“…in this random construction. Such results have been proved in [34, Proposition 3.1; 35]. However, the Steiner triple systems obtainable by Keevash's construction comprise a negligible proportion of the set of Steiner triple systems, and a somewhat more delicate approach is required to study a uniformly random Steiner triple system.…”

Almost all Steiner triple systems are almost resolvable

“…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]. It would be very interesting if one could substantially improve the error term 2 ; we imagine the correct order of magnitude is We imagine that actually, the number of (labelled) copies of should a.a.s.…”

Coloured and Directed Designs

“…'almost entirely random', in that the semi-random (nibble) construction of approximate designs by Rödl [23] can be completed to an actual design by an absorption process (Randomised Algebraic Construction in [10] or Iterative Absorption in [4]). In this vein, we mention the proof by Kwan [14] that almost all Steiner triple systems have perfect matchings, results on discrepancy of highdimensional permutations by Linial and Luria [16], and the existence of bounded degree coboundary expanders of every dimension by Lubotzky, Luria and Rosenthal [17]. These results suggest that the new results in [11] may create more fruitful connections with the theory of high-dimensional expanders and other topics in high-dimensional combinatorics.…”