Lower bounds on the sizes of defining sets in full $n$-Latin squares and full designs

TL;DR

This paper establishes asymptotically tight lower bounds on the sizes of defining sets for full $n$-Latin squares and full designs, improving previous results and solving an open problem about the proportion of blocks in full designs.

Contribution

It provides the first asymptotically optimal lower bounds on defining set sizes for full Latin squares and designs, confirming conjectures about their asymptotic behavior.

Findings

Any defining set for the full $n$-Latin square has size $n^3(1-o(1))$.

Any defining set for the full design $N(v,k)$ has size ${vrace k}(1-o(1))$.

These bounds are asymptotically optimal and resolve an open problem from 2009.

Abstract

The full $n$-Latin square is the $n\times n$ array with symbols $1,2,\dots ,n$ in each cell. In this paper we show, as part of a more general result, that any defining set for the full $n$-Latin square has size $n^3(1-o(1))$. The full design $N(v,k)$ is the unique simple design with parameters $(v,k,{v-2 \choose k-2})$; that is, the design consisting of all subsets of size $k$ from a set of size $v$. We show that any defining set for the full design $N(v,k)$ has size ${v\choose k}(1-o(1))$ (as $v-k$ becomes large). These results improve existing results and are asymptotically optimal. In particular, the latter result solves an open problem given in (Donovan, Lefevre, et al, 2009), in which it is conjectured that the proportion of blocks in the complement of a full design will asymptotically approach zero.