# Monochromatic paths in random tournaments

## TL;DR

This paper proves that in a randomly coloured tournament, there is likely a long monochromatic path, resolving a conjecture and providing bounds on the oriented size Ramsey number of directed paths.

## Contribution

It establishes a high-probability bound on monochromatic path length in random tournaments, confirming a conjecture and advancing understanding of oriented Ramsey theory.

## Key findings

- Monochromatic paths of length Ω(n / sqrt(log n)) exist with high probability.
- Resolves a conjecture by Ben-Eliezer, Krivelevich, and Sudakov.
- Provides nearly tight bounds on the oriented size Ramsey number.

## Abstract

We prove that, with high probability, any $2$-edge-colouring of a random tournament on $n$ vertices contains a monochromatic path of length $\Omega(n / \sqrt{\log n})$. This resolves a conjecture of Ben-Eliezer, Krivelevich and Sudakov and implies a nearly tight upper bound on the oriented size Ramsey number of a directed path.