Large unavoidable subtournaments

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 $n_0(k,\epsilon )\le {\epsilon }^{-O(k/{\epsilon }^2)}$ and they conjectured that this could be reduced to $n_0(k,\epsilon )\le {\epsilon }^{-O(k)}$. Here we prove this conjecture.

9 pages

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