  For a hypergraph G and a positive integer s, let \chi_{\ell} (G,s) be the
minimum value of l such that G is L-colorable from every list L with |L(v)|=l
for each v\in V(G) and |L(u)\cap L(v)|\leq s for all u, v\in e\in E(G). This
parameter was studied by Kratochv\'{i}l, Tuza and Voigt for various kinds of
graphs. Using randomized constructions we find the asymptotics of \chi_{\ell}
(G,s) for balanced complete multipartite graphs and for complete k-partite
k-uniform hypergraphs.
