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 V such that L(u) L(v) p + r for all edges uv E, do there exist q-element subsets C(v) L(v) with C(u) ∩ C(v) = 0 for all uv 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 {L(v)v V{, is closely related to Turán's problem on uniform hypergraphs.