Monochromatic trees in random tournaments

TL;DR

This paper proves that in random tournaments with two-edge coloring, one can find monochromatic copies of all oriented trees up to size proportional to n divided by the square root of log n, extending previous path results.

Contribution

It generalizes prior work from paths to all oriented trees of a certain size in random tournaments, establishing tight bounds.

Findings

Monochromatic copies of all oriented trees of size O(n / sqrt(log n)) exist with high probability.

The result extends previous path-specific findings to more complex tree structures.

The bounds are tight up to a constant factor.

Abstract

We prove that, with high probability, in every $2$-edge-colouring of the random tournament on $n$ vertices there is a monochromatic copy of every oriented tree of order $O (n / \sqrt{\log n})$. This generalises a result of the first, third and forth authors who proved the same statement for paths, and is tight up to a constant factor.