Year: 2016
Type: article
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.
Author: EOIN LONG
Institution: Tel Aviv University
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W1747816245