Latin squares with maximal partial transversals of many lengths

A partial transversal T of a Latin square Lis a set of entries of L in which each row, column and symbol is represented at most once. A partial transversal is maximal if it is not contained in a larger partial transversal. Any maximal partial transversal of a Latin square of order n has size at least (iverted right perpendicular)n/2(inverted left perpendicular) and at most n. We say that a Latin square is omniversal if it possesses a maximal partial transversal of all plausible sizes and is near-omniversal if it possesses a maximal partial transversal of all plausible sizes except one. Evans (2019) showed that omniversal Latin squares of order n exist for any odd n not equal 3. By extending this result, we show that an omniversal Latin square of order n exists if and only if n is not an element of {3, 4} and n not equal 2(mod 4). Furthermore, we show that near-omniversal Latin squares exist for all orders n equivalent to 2 (mod 4). Finally, we show that no non-trivial finite group has an omniversal Cayley table, and only 15 finite groups have a near-omniversal Cayley table. In fact, as n grows, Cayley tables of groups of order n miss a constant fraction of the plausible sizes of maximal partial transversals. In the course of proving this, we partially solve the following interesting problem in combinatorial group theory. Suppose that we have two finite subsets R, C subset of G of a group G such that vertical bar{rc : r is an element of R, c is an element of C}vertical bar = m. How large do vertical bar R vertical bar and vertical bar C vertical bar need to be (in terms of m) to be certain that R subset of xH and C subset of Hy for some subgroup H of order min G, and x, y is an element of G? (C) 2021 Elsevier Inc. All rights reserved.