TR19-048 Authors: Per Austrin, Amey Bhangale, Aditya Potukuchi

Publication: 2nd April 2019 12:46

Downloads: 729

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We reprove the results on the hardness of approximating hypergraph coloring using a different technique based on bounds on the size of extremal $t$-agreeing families of $[q]^n$. Specifically, using theorems of Frankl-Tokushige [FT99], Ahlswede-Khachatrian [AK98] and Frankl [F76] on the size of such families, we give simple and unified proofs of quasi NP-hardness of the following problems:

$\bullet$ coloring a $3$ colorable $4$-uniform hypergraph with $(\log n)^\delta$ many colors

$\bullet$ coloring a $3$ colorable $3$-uniform hypergraph with $\tilde{O}(\sqrt{\log \log n})$ many colors

$\bullet$ coloring a $2$ colorable $6$-uniform hypergraph with $(\log n)^\delta$ many colors

$\bullet$ coloring a $2$ colorable $4$-uniform hypergraph with $\tilde{O}(\sqrt{\log \log n})$ many colors

where $n$ is the number of vertices of the hypergraph and $\delta>0$ is a universal constant.