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Large Unavoidable Subtournaments

Part of: Extremal combinatorics Graph theory

Published online by Cambridge University Press:  21 June 2016

EOIN LONG
EOIN LONG*
Affiliation:
School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel (e-mail: [email protected])
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Abstract

Let Dk denote the tournament on 3k vertices consisting of three disjoint vertex classes V1, V2 and V3 of size k, each oriented as a transitive subtournament, and with edges directed from V1 to V2, from V2 to V3 and from V3 to V1. Fox and Sudakov proved that given a natural number k and ε > 0, there is n0(k, ε) such that every tournament of order n ⩾ n0(k,ε) which is ε-far from being transitive contains Dk as a subtournament. Their proof showed that $n_0(k,\epsilon ) \leq \epsilon ^{-O(k/\epsilon ^2)}$ and they conjectured that this could be reduced to n0(k, ε) ⩽ ε−O(k). Here we prove this conjecture.

MSC classification

Primary: 05D10: Ramsey theory
Secondary: 05C20: Directed graphs (digraphs), tournaments
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 26 , Issue 1 , January 2017 , pp. 68 - 77
Copyright
Copyright © Cambridge University Press 2016 

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