Extending the Gyárfás-Sumner conjecture

Say a set H of graphs is heroic if there exists k such that every graph containing no member of H as an induced subgraph has cochromatic number at most k. (The cochromatic number of G is the minimum number of stable sets and cliques with union V(G) Assuming an old conjecture of Gyárfás and Sumner, we give a complete characterization of the finite heroic sets.This is a consequence of the following. Say a graph is k-split if its vertex set is the union of two sets A, B, where A has clique number at most k and B has stability number at most k. For every graph H₁ that is a disjoint union of cliques, and every complete multipartite graph H₂, there exists k such that every graph containing neither of H₁, H₂ as an induced subgraph is k-split.This in turn is a consequence of a bound on the maximum number of vertices in any graph that is minimal not k-split, a result first proved by Gyárfás [5] and for which we give a short proof.