Excluding paths and antipaths

The Erdős-Hajnal conjecture states that for every graph H, there exists a constant δ(H)>0, such that if a graph G has no induced subgraph isomorphic to H, then G contains a clique or a stable set of size at least |V (G)|^δ(H). This conjecture is still open. We consider a variant of the conjecture, where instead of excluding H as an induced subgraph, both H and H^c are excluded. We prove this modified conjecture for the case when H is the five-edge path. Our second main result is an asymmetric version of this: we prove that for every graph G such that G contains no induced six-edge path, and G^c contains no induced four-edge path, G contains a polynomial-size clique or stable set.