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Erd\"{o}s-Hajnal conjecture states that for every undirected graph $H$ thereexists $ \epsilon(H) > 0 $ such that every undirected graph on $ n $ verticesthat does not contain $H$ as an induced subgraph contains a clique or a stableset of size at least $ n^{\epsilon(H)} $. This conjecture has a directedequivalent version stating that for every tournament $H$ there exists $\epsilon(H) > 0 $ such that every $H-$free $n-$vertex tournament $T$ contains atransitive subtournament of order at least $ n^{\epsilon(H)} $. This conjectureis known to hold for a few infinite families of tournaments. In this paper weconstruct two new infinite families of tournaments - the family of so-calledgalaxies with spiders and the family of so-called asterisms, and we prove thecorrectness of the conjecture for these two families.

Erd\"{o}s-Hajnal Conjecture for New Infinite Families of Tournaments