  A rainbow $q$-coloring of a $k$-uniform hypergraph is a $q$-coloring of the
vertex set such that every hyperedge contains all $q$ colors.
  We prove that given a rainbow $(k - 2\lfloor \sqrt{k}\rfloor)$-colorable
$k$-uniform hypergraph, it is NP-hard to find a normal $2$-coloring.
Previously, this was only known for rainbow $\lfloor k/2 \rfloor$-colorable
hypergraphs (Guruswami and Lee, SODA 2015).
  We also study a generalization which we call rainbow $(q, p)$-coloring,
defined as a coloring using $q$ colors such that every hyperedge contains at
least $p$ colors. We prove that given a rainbow $(k - \lfloor \sqrt{kc}
\rfloor, k- \lfloor3\sqrt{kc} \rfloor)$-colorable $k$ uniform hypergraph, it is
NP-hard to find a normal $c$-coloring for any $c = o(k)$.
  The proof of our second result relies on two combinatorial theorems. One of
the theorems was proved by Sarkaria (J. Comb. Theory. 1990) using topological
methods and the other theorem we prove using a generalized Borsuk-Ulam theorem.
