Large unavoidable subtournaments

TL;DR

This paper proves a conjecture that a large tournament far from being transitive necessarily contains a specific structured subtournament, improving the bounds on the size needed for such containment.

Contribution

The paper confirms that the size threshold for containing the structured subtournament $D_k$ can be bounded by a polynomial in $rac{1}{ ext{epsilon}}$, refining previous exponential bounds.

Findings

Confirmed the conjecture on the size bound for containing $D_k$

Improved the upper bound from exponential to polynomial in $rac{1}{ ext{epsilon}}$

Strengthened understanding of structure in large tournaments

Abstract

Let $D_k$ denote the tournament on $3k$ vertices consisting of three disjoint vertex classes $V_1, V_2$ and $V_3$ of size $k$, each of which is oriented as a transitive subtournament, and with edges directed from $V_1$ to $V_2$, from $V_2$ to $V_3$ and from $V_3$ to $V_1$. Fox and Sudakov proved that given a natural number $k$ and $\epsilon > 0$ there is $n_0(k,\epsilon )$ such that every tournament of order $n_0(k,\epsilon )$ which is $\epsilon $-far from being transitive contains $D_k$ 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 $n_0(k,\epsilon ) \leq \epsilon ^{-O(k)}$. Here we prove this conjecture.