Abstract
An ordered graph is a pair $\mathcal{G}=(G,\prec)$ where $G$ is a graph and $\prec$ is a total ordering of its vertices.
 The ordered Ramsey number $\overline{R}(\mathcal{G})$ is the minimum number $N$ such that every ordered complete graph with $N$ vertices and with edges colored by two colors contains a monochromatic copy of $\mathcal{G}$.
 In contrast with the case of unordered graphs, we show that there are arbitrarily large ordered matchings $\mathcal{M}_n$ on $n$ vertices for which $\overline{R}(\mathcal{M}_n)$ is superpolynomial in $n$. This implies that ordered Ramsey numbers of the same graph can grow superpolynomially in the size of the graph in one ordering and remain linear in another ordering.
 We also prove that the ordered Ramsey number $\overline{R}(\mathcal{G})$ is polynomial in the number of vertices of $\mathcal{G}$ if the bandwidth of $\mathcal{G}$ is constant or if $\mathcal{G}$ is an ordered graph of constant degeneracy and constant interval chromatic number.
 The first result gives a positive answer to a question of Conlon, Fox, Lee, and Sudakov.
 For a few special classes of ordered paths, stars or matchings, we give asymptotically tight bounds on their ordered Ramsey numbers. For so-called monotone cycles we compute their ordered Ramsey numbers exactly. This result implies exact formulas for geometric Ramsey numbers of cycles introduced by Károlyi, Pach, Tóth, and Valtr.
Highlights
Ramsey’s theorem [35] states that for every given graph G, every sufficiently large complete graph with edges colored by a constant number of colors contains a monochromatic copy of G
The main goal of this paper is to understand the effects of different vertex orderings on the ordered Ramsey number of a given graph, and to compare the ordered and unordered Ramsey numbers
In Subsection 2.2, we show an ordering (Pn, alt) of the path Pn whose ordered Ramsey number is linear in n
Summary
Ramsey’s theorem [35] states that for every given graph G, every sufficiently large complete graph with edges colored by a constant number of colors contains a monochromatic copy of G. Since every graph on n vertices is contained in Kn, we can consider the following generalization of Ramsey numbers. Gc) the smallest number N such that every c-coloring of KN contains, for some i, a monochromatic copy of Gi in color i as an ordered subgraph. There is only one ordered complete graph on n vertices, which we denote by Kn. for arbitrary positive integers c, r1, . To study the asymptotic growth of ordered Ramsey numbers, we introduce ordering schemes for some classes of graphs. We know that the ordered Ramsey number Rk(H) of an arbitrary ordered k-uniform hypergraph H is finite
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