Lower Bounds on the Sizes of Defining Sets in Full n-Latin Squares and Full Designs

Abstract

The full n-Latin square is the $n\times n$ array with symbols $1,2,\ldots ,n$ 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 $(v,k,{v-2 \atopwithdelims ()k-2})$; 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 ${v\atopwithdelims ()k}(1-o(1))$ (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 et al. (Graphs Comb 25:825–839, 2009), in which it is conjectured that the proportion of blocks in the complement of a full design will asymptotically approach zero.