On the Problem of Erdős and Hajnal in the Case of List Colorings

We deal with the classical problem of Erdős and Hajnal in hypergraph theory and its generalization concerning the list colorings of hypergraphs. Let m(n,k) (mlist(n,k)) denote the minimum number of edges in an n-uniform hypergraph with chromatic (list chromatic) number k+1. We obtained some new lower bounds for m(n,k) and mlist(n,k) which improved previous results for some values of parameters n and k.