Simplified inpproximability of hypergraph coloring via t-agreeing families

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 $(\log n)^\delta$manycolors$\bullet$coloringa$3$colorable$3$-uniformhypergraphwith$\tilde{O}(\sqrt{\log \log n})$manycolors$\bullet$coloringa$2$colorable$6$-uniformhypergraphwith$(\log n)^\delta$manycolors$\bullet$coloringa$2$colorable$4$-uniformhypergraphwith$\tilde{O}(\sqrt{\log \log n})$ many colors where $n$ is the number of vertices of the hypergraph and $\delta >0$ is a universal constant.