Monochromatic paths in random tournaments

Abstract

We prove that, with high probability, any 2-edge-colouring of a random tournament on n vertices contains a monochromatic path of length Ω(n/ √ 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.