  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.
