A note on hypergraph colorings

Let $t\geqslant 2$ and $s\geqslant 1$ be two integers. Define a $(t,s)$-coloring of a hypergraph to be a coloring of its vertices using $t$ colors such that each color appears on each edge at least $s$ times. In this note, we provide a sufficient condition for the existence of a $(t,s)$-coloring of a hypergraph by using the symmetric lopsided version of Lov\'asz Local Lemma. Our result generalizes several known results on hypergraph colorings.