Erd"{o}s-Hajnal Conjecture for Flotilla-Galaxy Tournaments (original) (raw)
2020, arXiv: Combinatorics
Erd\"{o}s-Hajnal conjecture states that for every undirected graph HHH there exists $ \epsilon(H) > 0 $ such that every undirected graph on $ n $ vertices that does not contain HHH as an induced subgraph contains a clique or a stable set of size at least $ n^{\epsilon(H)} .Thisconjecturehasadirectedequivalentversionstatingthatforeverytournament. This conjecture has a directed equivalent version stating that for every tournament .ThisconjecturehasadirectedequivalentversionstatingthatforeverytournamentH$ there exists $ \epsilon(H) > 0 $ such that every H−H-H−free n−n-n−vertex tournament TTT contains a transitive subtournament of order at least $ n^{\epsilon(H)} .Thisconjectureisprovedwhen. This conjecture is proved when .ThisconjectureisprovedwhenH$ is a galaxy or a constellation and for all five$-$vertex tournaments and for all six$-$vertex tournaments except one. In this paper we prove the correctness of the conjecture for any flotilla-galaxy tournament. This generalizes results some previous results.
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