An { \it omnimosaic} $O(n,k,a)$ is defined to be an $n\times n$ matrix, with entries from the set ${ \cal A}=\{ 1,2,\...,a\}$, that contains, as a submatrix, each of the $a^{ k^2}$ $k\times k$ matrices over ${ \cal A}$. We provide constructions of omnimosaics and show that for fixed $a$ the smallest possible size $\omega(k,a)$ of an $O(n,k,a)$ omnimosaic satisfies \[\frac{ ka^{ k/2}}{ e}\le \omega(k,a)\le \frac{ ka^{ k/2}}{ e}(1+o(1))\] for a well-specified function $o(1)$ that tends to zero as $k\to\infty$.
Categories: math.CO math.PR