The size Ramsey number of a directed path

Authors:
Ido Ben-Eliezer;Michael Krivelevich;Benny Sudakov
Affiliations:
School of Computer Science, Raymond and Beverly Sackler Faculy of Exact Sciences, Tel Aviv University, Tel Aviv 69978, Israel;School of Mathematical Sciences, Raymond and Beverly Sackler Faculy of Exact Sciences, Tel Aviv University, Tel Aviv 69978, Israel;Department of Mathematics, UCLA, Los Angeles, CA 90095, United States
Venue:
Journal of Combinatorial Theory Series B
Year:
2012

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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 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 c"1=c"1(q),c"2 such thatc"1(q)n^2^q(logn)^1^/^q(loglogn)^(^q^+^2^)^/^q=,q+1)=,q) for general directed graphs with q=3, extending previous results.