Forbidding Couples of Tournaments and the Erd\"{o}s-Hajnal Conjecture

TL;DR

This paper investigates the Erdős-Hajnal conjecture for specific classes of tournaments, establishing partial results and bounds for certain families like super nebulas and super triangular galaxies, advancing understanding in directed graph theory.

Contribution

The paper constructs infinite families of tournaments where the Erdős-Hajnal conjecture remains open and proves new bounds for tournaments avoiding these families and K6.

Findings

Proved existence of polynomial-sized transitive subtournaments in certain H-free tournaments.

Established bounds for super nebula and super triangular galaxy families.

Extended results to include K6 and other tournament classes.

Abstract

A celebrated unresolved conjecture of Erd\"{o}s and Hajnal states that for every undirected graph $H$ there exists $ \epsilon(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^{\epsilon(H)} $. This conjecture has a directed equivalent version stating that for every tournament $H$ there exists $ \epsilon(H) > 0 $ such that every $H-$free $n-$vertex tournament $T$ contains a transitive subtournament of size at least $ n^{\epsilon(H)} $. Recently the conjecture was proved for all six-vertex tournaments, except $K_{6}$. In this paper we construct two infinite families of tournaments for which the conjecture is still open for infinitely many tournaments in these two families $-$ the family of so-called super nebulas and the family of so-called super triangular galaxies. We prove that for every super nebula $H_{1}$ and every $\Delta$galaxy $H_{2}$ there exist $\epsilon(H_{1},H_{2})$ such that every $\lbrace H_{1},H_{2}\rbrace$$-$free tournament $T$ contains a transitive subtournament of size at least $\mid$$T$$\mid^{\epsilon(H_{1},H_{2})}$. We also prove that for every central triangular galaxy $H$ there exist $\epsilon(K_{6},H)$ such that every $\lbrace K_{6},H\rbrace$$-$free tournament $T$ contains a transitive subtournament of size at least $\mid$$T$$\mid^{\epsilon(K_{6},H)}$. And we give an extension of our results.