On edge-ordered Ramsey numbers

An edge-ordered graph is a graph with a linear ordering of its edges. Two edge-ordered graphs are equivalent if their is an isomorphism between them preserving the ordering of the edges. The edge-ordered Ramsey number $r_{edge}(H; q)$ of an edge-ordered graph $H$ is the smallest $N$ such that there exists an edge-ordered graph $G$ on $N$ vertices such that, for every $q$-coloring of the edges of $G$, there is a monochromatic subgraph of $G$ equivalent to $H$. Recently, Balko and Vizer announced that $r_{edge}(H;q)$ exists. However, their proof uses the Graham-Rothschild theorem and consequently gives an enormous upper bound on these numbers. We give a new proof giving a much better bound. We prove that for every edge-ordered graph $H$ on $n$ vertices, we have $r_{edge}(H;q) \leq 2^{c^qn^{2q-2}\log^q n}$, where $c$ is an absolute constant. We also explore the edge-ordered Ramsey number of sparser graphs and prove a polynomial bound for edge-ordered graphs of bounded degeneracy. We also prove a strengthening for edge-labeled graphs, graphs where every edge is given a label and the labels do not necessary have an ordering.