Finite Induced Graph Ramsey Theory: On Partitions of Subgraphs

For given finite (unordered) graphs G and H, we examine the existence of a Ramsey graph F for which the strong Ramsey arrow F → (G)Hr holds. We concentrate on the situation when H is not a complete graph. The set of graphs G for which there exists an F satisfying F → (G)P22 (P2 is a path on three vertices) is found to be the union of the set of chordal comparability graphs together with the set of convex graphs.