Complexity of choosing subsets from color sets

We raise and investigate the algorithmic complexity of the following problem. Given a graph G = (V; E) and p-element sets L(v) for its vertices v 2 V such that jL(u) L(v)j p + r for all edges uv 2 E, do there exist q-element subsets C (v) L(v) with C (u) \ C (v) = ; for all uv 2 E ? Here p; q; r are positive integers, p q and p + r 2q. We characterize precisely which triples (p; q; r) admit a polynomial solution, and for which ones the problem is NP-complete. Moreover, it is shown that for some restricted ranges of p and r with respect to q, the existence of subsets C (v) L(v) for every collection fL(v) j v 2 V g is closely related to Turr an's problem on uniform hypergraphs.