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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, thatany defining set for the full $n$-Latin square has size $n^3(1-o(1))$. The fulldesign $N(v,k)$ is the unique simple design with parameters $(v,k,{v-2 \choosek-2})$; that is, the design consisting of all subsets of size $k$ from a set ofsize $v$. 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 existingresults and are asymptotically optimal. In particular, the latter result solvesan open problem given in (Donovan, Lefevre, et al, 2009), in which it isconjectured that the proportion of blocks in the complement of a full designwill asymptotically approach zero.

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