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�վ��ٱ���:��A Sudoku Baranyai's Theorem

Abstract: Motivated by constructing higher dimensional Sudokus we generalize the famous Baranyai's theorem. Let$n=\prod_{i=1}^d a_i$. Suppose that an $n\times\dots \times n$ ($d$ times) array $L$ is partitioned into$n/a_1\times\dots\times n/a_d$ sub‐arrays (called blocks). Can we color the $n^d$ cells of $L$ with $n^{d21}$ colors so that each layer (obtained by fixing one coordinate) and each $n/a_1\times\dots\times n/a_d$block contains each color exactly once? We generalize the well‐know theorem of Baranyai to answer this queestion. The case $d=2 a_1=a_2=3$ corresponds to the usual Sudoku. We also provide finite fieldconstruction of various related objects. This is joint work with Sho Suda.