The size Ramsey number of a directed path

Abstract

Given a graph H, the size Ramsey number re(H,q) is the minimal number m for which there is a graph G with m edges such that every q-coloring of E(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⩾1 there are constants c1=c1(q),c2 such thatc1(q)n2q(logn)1/q(loglogn)(q+2)/q⩽re(Pn→,q+1)⩽c2n2q(logn)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 on the number of colors, as opposed to the undirected case.Our approach also gives tight bounds on re(Pn→,q) for general directed graphs with q⩾3, extending previous results.