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The oriented size Ramsey number of directed paths
- Publication Year :
- 2017
-
Abstract
- 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 $r(P_n)$.<br />Comment: 6 pages
- Subjects :
- Mathematics - Combinatorics
Subjects
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.1712.02403
- Document Type :
- Working Paper