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Omnimosaics

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 $ω(k,a)$ of an $O(n,k,a)$ omnimosaic satisfies \[\frac{ka^{k/2}}{e}\le ω(k,a)\le \frac{ka^{k/2}}{e}(1+o(1))\] for a well-specified function $o(1)$ that tends to zero as $k\to\infty$.