The size Ramsey number of a directed path

TL;DR

This paper establishes nearly tight bounds on the size Ramsey number of directed paths in oriented graphs, revealing its polynomial dependence on the number of colors and its asymptotic difference from general directed graphs.

Contribution

It provides the first nearly tight bounds for the size Ramsey number of directed paths in oriented graphs for any fixed number of colors.

Findings

Bounds are nearly tight for all fixed q.

Size Ramsey number in oriented graphs is larger than in general directed graphs.

Polynomial dependence on the number of colors is shown.

Abstract

Given a graph $H$, the size Ramsey number $r_e(H,q)$ is the minimal number $m$ for which there is a graph $G$ with $m$ edges such that every $q$-coloring of $G$ contains a monochromatic copy of $H$. We study the size Ramsey number of the directed path of length $n$ in oriented graphs, where no antiparallel edges are allowed. We give nearly tight bounds for every fixed number of colors, showing that for every $q\geq 1 $ there are constants $c_1 = c_1(q),c_2$ such that $$\frac{c_1(q) n^{2q}(\log n)^{1/q}}{(\log\log n)^{(q+2)/q}} \leq r_e(\overrightarrow{P_n},q+1) \leq c_2 n^{2q}(\log {n})^2.$$ Our results show that the path size Ramsey number in oriented graphs is asymptotically larger than the path size Ramsey number in general directed graphs. Moreover, the size Ramsey number of a directed path is polynomially dependent in the number of colors, as opposed to the undirected case. Our approach also gives tight bounds on $r_e(\overrightarrow{P_n},q)$ for general directed graphs with $q \geq 3$, extending previous results.