Tournaments with near-linear transitive subsets

Krzysztof Choromanski, Maria Chudnovsky, Paul Seymour

Research output: Contribution to journalArticle

6 Scopus citations

Abstract

Let H be a tournament, and let ε≥0 be a real number. We call ε an "Erdos-Hajnal coefficient" for H if there exists c>0 such that in every tournament G not containing H as a subtournament, there is a transitive subset of cardinality at least c|V(G)|ε. The Erdos-Hajnal conjecture asserts, in one form, that every tournament H has a positive Erdos-Hajnal coefficient. This remains open, but recently the tournaments with Erdos-Hajnal coefficient 1 were completely characterized. In this paper we provide an analogous theorem for tournaments that have an Erdos-Hajnal coefficient larger than 5/6; we give a construction for them all, and we prove that for any such tournament H there are numbers c, d such that, if a tournament G with |V(G)|>1 does not contain H as a subtournament, then V(G) can be partitioned into at most c(log(|V(G)|))d transitive subsets.

Original languageEnglish (US)
Pages (from-to)228-249
Number of pages22
JournalJournal of Combinatorial Theory. Series B
Volume109
DOIs
StatePublished - Nov 1 2014

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics
  • Computational Theory and Mathematics

Keywords

  • The erdos-hajnal conjecture
  • Tournaments

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