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A conjecture of Alon, Pach and Solymosi, which is equivalent to thecelebrated Erd\H{o}s-Hajnal Conjecture, states that for every tournament $S$there exists $\epsilon(S)>0$ such that if $T$ is an $n$-vertex tournament thatdoes not contains $S$ as a subtournament, then $T$ contains a transitivesubtournament on at least $n^{\epsilon(S)}$ vertices. Let $C_5$ be the uniquefive-vertex tournament where every vertex has two inneighbors and twooutneighbors. The Alon-Pach-Solymosi conjecture is known to be true for thecase when $S=C_5$. Here we prove a strengthening of this result, showing thatin every tournament $T$ with no subtorunament isomorphic to $C_5$ there existdisjoint vertex subsets $A$ and $B$, each containing a linear proportion of thevertices of $T$, and such that every vertex of $A$ is adjacent to every vertexof $B$.

Tournaments and the Strong Erd\H{o}s-Hajnal Property