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 $(log n)^delta$ 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.