Large cliques or stable sets in graphs with no four-edge path and no five-edge path in the complement

Erdős and Hajnal [4] conjectured that, for any graph H, every graph on n vertices that does not have H as an induced subgraph contains a clique or a stable set of size n for some ε(H) > 0. The conjecture is known to be true for graphs H with |V (H)| ≤ 4. One of the two remaining open cases on five vertices is the case where H is a four-edge path, the other case being a cycle of length five. In this paper we prove that every graph on n vertices that does not contain a four-edge-path or the complement of a five-edge-path as an induced subgraph contains either a clique or a stable set of size at least n. ∗Partially supported by NSF grant DMS-0758364. †This research was performed while the author was at Columbia University.