  The celebrated Erdos-Hajnal conjecture states that for every $n$-vertex
undirected graph $H$ there exists $\eps(H)>0$ such that every graph $G$ that
does not contain $H$ as an induced subgraph contains a clique or an independent
set of size at least $n^{\eps(H)}$. A weaker version of the conjecture states
that the polynomial-size clique/independent set phenomenon occurs if one
excludes both $H$ and its complement $H^{\compl}$. We show that the weaker
conjecture holds if $H$ is any path with a pendant edge at its third vertex;
thus we give a new infinite family of graphs for which the conjecture holds.
