On the Erdős–Hajnal conjecture for six-vertex tournaments

Eli Berger, Krzysztof Choromanski, Maria Chudnovsky

Research output: Contribution to journalArticlepeer-review

Abstract

A celebrated unresolved conjecture of Erdős and Hajnal states that for every undirected graph H there exists ϵ(H)>0 such that every undirected graph on n vertices that does not contain H as an induced subgraph contains a clique or stable set of size at least nϵ(H). The conjecture has a directed equivalent version stating that for every tournament H there exists ϵ(H)>0 such that every H-free n-vertex tournament T contains a transitive subtournament of order at least nϵ(H). We say that a tournament is prime if it does not have nontrivial homogeneous sets. So far the conjecture was proved only for some specific families of prime tournaments (Berger et al., 2014; Choromanski, 2015 [3]) and tournaments constructed according to the so-called substitution procedure (Alon et al., 2001). In particular, recently the conjecture was proved for all five-vertex tournaments (Berger et al. 2014), but the question about the correctness of the conjecture for all six-vertex tournaments remained open. In this paper we prove that all but at most one six-vertex tournament satisfy the Erdős–Hajnal conjecture. That reduces the six-vertex case to a single tournament.

Original languageEnglish (US)
Pages (from-to)113-122
Number of pages10
JournalEuropean Journal of Combinatorics
Volume75
DOIs
StatePublished - Jan 2019

All Science Journal Classification (ASJC) codes

  • Discrete Mathematics and Combinatorics

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