TY - JOUR

T1 - The size Ramsey number of a directed path

AU - Ben-Eliezer, Ido

AU - Krivelevich, Michael

AU - Sudakov, Benny

N1 - Funding Information: E-mail addresses: idobene@post.tau.ac.il (I. Ben-Eliezer), krivelev@post.tau.ac.il (M. Krivelevich), bsudakov@math.ucla.edu (B. Sudakov). 1 Research supported in part by the Dan David fellowship for PhD students and by an ERC advanced grant. 2 Research supported in part by USA–Israel BSF grant 2006322, and by grant 1063/08 from the Israel Science Foundation. 3 Research supported in part by NSF grant DMS-1101185, NSF CAREER award DMS-0812005 and by USA–Israeli BSF grant.

PY - 2012/5

Y1 - 2012/5

N2 - 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 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.

AB - 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 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.

KW - Monochromatic directed paths

KW - Ramsey theory

KW - Size Ramsey number

UR - http://www.scopus.com/inward/record.url?scp=84857637430&partnerID=8YFLogxK

U2 - https://doi.org/10.1016/j.jctb.2011.10.002

DO - https://doi.org/10.1016/j.jctb.2011.10.002

M3 - مقالة

SN - 0095-8956

VL - 102

SP - 743

EP - 755

JO - Journal of Combinatorial Theory. Series B

JF - Journal of Combinatorial Theory. Series B

IS - 3

ER -