The oriented size Ramsey number of directed paths

An oriented graph is a directed graph with no bi-directed edges, i.e. if $xy$ is an edge then $yx$ is not an edge. The oriented size Ramsey number of an oriented graph $H$, denoted by $r(H)$, is the minimum $m$ for which there exists an oriented graph $G$ with $m$ edges, such that every $2$-colouring of $G$ contains a monochromatic copy of $H$. In this paper we prove that the oriented size Ramsey number of the directed paths on $n$ vertices satisfies $r(P_n) = \Omega(n^2 \log n)$. This improves a lower bound by Ben-Eliezer, Krivelevich and Sudakov. It also matches an upper bound by Buci\'c and the authors, thus establishing an asymptotically tight bound on $r(P_n)$. We also discuss how our methods can be used to improve the best known lower bound of the $k$-colour version of $r(P_n)$.
Comment: 6 pages