  A pure pair in a tournament $G$ is an ordered pair $(A,B)$ of disjoint
subsets of $V(G)$ such that every vertex in $B$ is adjacent from every vertex
in $A$. Which tournaments $H$ have the property that if $G$ is a tournament not
containing $H$ as a subtournament, and $|G|>1$, there is a pure pair $(A,B)$ in
$G$ with $|A|,|B|\ge c|G|$, where $c>0$ is a constant independent of $G$? Let
us say that such a tournament $H$ has the strong EH-property.
  As far as we know, it might be that a tournament $H$ has this property if and
only if its vertex set has a linear ordering in which its backedges form a
forest. Certainly this condition is necessary, but we are far from proving
sufficiency. We make a small step in this direction, showing that if a
tournament can be ordered with at most three backedges then it has the strong
EH-property (except for one case, that we could not decide). In particular,
every tournament with at most six vertices has the property, except for three
that we could not decide. We also give a seven-vertex tournament that does not
have the strong EH-property.
  This is related to the Erdos-Hajnal conjecture, which in one form says that
for every tournament $H$ there exists $\tau>0$ such that every tournament $G$
not containing $H$ as a subtournament has a transitive subtournament of
cardinality at least $|G|^\tau$. Let us say that a tournament $H$ satisfying
this has the EH-property. It is known that every tournament with the strong
EH-property also has the EH-property; so our result extends work by Berger,
Choromanski and Chudnovsky, who proved that every tournament with at most six
vertices has the EH-property, except for one that they did not decide.
