Coloring cross-intersecting families

Intersecting and cross-intersecting families usually appear in extremal combinatorics in the vein of the Erdos-Ko-Rado theorem. On the other hand, P. Erdos and L. Lovasz posed problems on coloring intersecting families as a restriction of classical hypergraph coloring problems to a special class of hypergraphs. We deal with the mentioned coloring problems stated for cross-intersecting families.