A celebrated unresolved conjecture of Erdös and Hajnal states that for every undirected graph H there exists (H) > 0 such that every undirected graph on n vertices that does not contain H as an induced subgraph contains a clique or a stable set of size at least n (H) . The conjecture has a directed equivalent version stating that for every tournament H there exists (H) > 0 such that every H−free n−vertex tournament T contains a transitive subtournament of order at least n (H) . Both the directed and the undirected versions of the conjecture are known to be true for small graphs (tournaments). So far the conjecture was proved only for some specific families of prime tournaments (Berger et al., 2014 [2]; Choromanski, 2015 [3]), tournaments constructed according to the so−called substitution procedure (Alon et al., 2001 [1]) allowing to build bigger graphs, and for all five−vertex tournaments (Berger et al. 2014 [2]). Recently the conjecture was proved for all six−vertex tournament, with one exception (Berger et al. 2018 [5]), but the question about the correctness of the conjecture for all seven−vertex tournaments remained open. In this paper we prove the correctness of the conjecture for several seven−vertex tournaments.

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Erdös-Hajnal conjecture states that for every undirected graph H there exists (H) > 0 such that every undirected graph on n vertices that does not contain H as an induced subgraph contains a clique or a stable set of size at least n (H) . This conjecture has a directed equivalent version stating that for every tournament H there exists (H) > 0 such that every H−free n−vertex tournament T contains a transitive subtournament of order at least n (H) . This conjecture is proved when H is a galaxy [2] or a constellation [5] and for all five−vertex tournaments [2] and for all six−vertex tournaments except K 6 [4]. In this paper we prove the correctness of the conjecture for any flotilla-galaxy tournament. This generalizes results in [2] and the results of [4]. IntroductionLet G be an undirected graph. We denote by V (G) the set of its vertices and by E(G) the set of its edges. We call |G| = |V (G)| the size of G. Let X ⊆ V (G), the subgraph of G induced by X is denoted by G|X, that is the graph with vertex set X, in which x, y ∈ X are adjacent if and only if they are adjacent in G. A clique in G is a set of pairwise adjacent vertices and a stable set in G is a set of pairwise nonadjacent vertices. For an undirected graph H, we say thatand such that for every (x, y) ∈ E we must have (y, x) / ∈ E, in particular if (x, y) ∈ E then x = y. E is the arc set