List colorings of $k$-partite $k$-graphs

A $k$-uniform hypergraph (or $k$-graph) $H=(V,E)$ is $k$-partite if $V$ can be partitioned into $k$ sets $V_1,\dots ,V_k$ such that each edge in $E$ contains precisely one vertex from each $V_i$. In this note, we consider list colorings for such hypergraphs. We show that for any $\epsilon >0$ if each vertex $v\in V(H)$ is assigned a list of size $\mathrm{\mid }L(v)\mathrm{\mid }\ge {((k-1+\epsilon )\mathrm{\Delta }\mathrm{/}\log \mathrm{\Delta })}^{1\mathrm{/}(k-1)}$, then $H$ admits a proper $L$-coloring, provided $\mathrm{\Delta }$ is sufficiently large. Up to a constant factor, this matches the bound on the chromatic number of simple $k$-graphs shown by Frieze and Mubayi, and that on the list chromatic number of triangle free $k$-graphs shown by Li and Postle. Our results hold in the more general setting of ``color-degree’’ as has been considered for graphs. Furthermore, we establish a number of asymmetric statements matching results of Alon, Cambie, and Kang for bipartite graphs.