An edge‐ordered graph is a graph with a linear ordering of its edges. Two edge‐ordered graphs are equivalent if there is an isomorphism between them preserving the edge‐ordering. 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 proved that r_edge(H; q) exists, but their proof gave enormous upper bounds on these numbers. We give a new proof with a much better bound, showing there exists a constant c such that for every edge‐ordered graph H on n vertices. We also prove a polynomial bound for the edge‐ordered Ramsey number of graphs of bounded degeneracy. Finally, we prove a strengthening for graphs where every edge has a label and the labels do not necessarily have an ordering.

中文翻译：

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在边缘排序的Ramsey数上

边有序图是具有边的线性排序的图。如果两个边缘有序图之间保持同构关系，则两个边缘有序图是等效的。边有序图H的边有序Ramsey数r _edge（H ;  q）是最小的N，因此在N个顶点上存在边有序图G，这样，对于每G个边的q着色，存在的单色子图ģ相当于ħ。最近，Balko和Vizer证明了r _edge（高; q）存在，但是他们的证明为这些数字提供了巨大的上限。我们给出了一个具有更好边界的新证明，表明存在一个常数c，对于n个顶点上的每个边序图H。我们还证明了有界退化的图的边序Ramsey数的多项式界。最后，我们证明了对于图的增强，其中每个边都有标签，标签不一定有序。