List colorings of multipartite hypergraphs

Let chi(l)(G) denote the list chromatic number of the r-uniform hypergraph G. Extending a result of Alon for graphs, Saxton and the second author used the method of containers to prove that, if G is simple and d-regular, then chi l(G)>=(1/(r-1)+o(1))logrd. To see how close this inequality is to best possible, we examine chi(l)(G) when G is a random r-partite hypergraph with n vertices in each class. The value when r = 2 was determined by Alon and Krivelevich; here we show that chi l(G)=(g(r,alpha)+o(1))logrd almost surely, where d is the expected average degree of G and alpha=lognd. The function g(r,alpha) is defined in terms of "preference orders" and can be determined fairly explicitly. This is enough to show that the container method gives an optimal lower bound on chi(l)(G) for r = 2 and r = 3, but, perhaps surprisingly, apparently not for r >= 4.