  An equivalent directed version of the celebrated unresolved conjecture of
Erdos and Hajnal proposed by Alon et al. states that for every tournament H
there exists epsilon(H) > 0 such that every H-free n-vertex tournament T
contains a transitive subtournament of order at least n^(epsilon(H)). A
tournament H has the strong EH-property if there exists c > 0 such that for
every H-free tournament T with |T| > 1, there exist disjoint vertex subsets A
and B, each of cardinality at least c|T| and every vertex of A is adjacent to
every vertex of B. Berger et al. proved that the unique five-vertex tournament
denoted by C5, where every vertex has two inneighbors and two outneighbors has
the strong EH-property. It is known that every tournament with the strong
EH-property also has the EH-property. In this paper we construct an infinite
class of tournaments - the so-called spiral galaxies and we prove that every
spiral galaxy has the strong EH-property.
