  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.
