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arXiv · 1410.5292

Ordered Ramsey numbers

David Conlon·Jacob Fox·Choongbum Lee·Benny Sudakov
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Abstract

Given a labeled graph $H$ with vertex set $\{1, 2,\ldots,n\}$, the ordered Ramsey number $r_<(H)$ is the minimum $N$ such that every two-coloring of the edges of the complete graph on $\{1, 2, \ldots,N\}$ contains a copy of $H$ with vertices appearing in the same order as in $H$. The ordered Ramsey number of a labeled graph $H$ is at least the Ramsey number $r(H)$ and the two coincide for complete graphs. However, we prove that even for matchings there are labelings where the ordered Ramsey number is superpolynomial in the number of vertices. Among other results, we also prove a general upper bound on ordered Ramsey numbers which implies that there exists a constant $c$ such that $r_<(H) \leq r(H)^{c \log^2 n}$ for any labeled graph $H$ on vertex set $\{1,2, \dots, n\}$.

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David Conlon, Jacob Fox, Choongbum Lee, Benny Sudakov. 2014-10-20. Ordered Ramsey numbers. https://arxiv.org/abs/1410.5292

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