An ordered graph is a graph equipped with a linear ordering of its vertex set. A pair of ordered graphs is Ramsey finite if it has only finitely many minimal ordered Ramsey graphs and Ramsey infinite otherwise. Here an ordered graph F is an ordered Ramsey graph of a pair (H,H') of ordered graphs if for any coloring of the edges of F in colors red and blue there is either a copy of H with all edges colored red or a copy of H' with all edges colored blue. Such an ordered Ramsey graph is minimal if neither of its proper subgraphs is an ordered Ramsey graph of (H,H'). If H=H' then H itself is called Ramsey finite. We show that a connected ordered graph is Ramsey finite if and only if it is a star with center being the first or the last vertex in the linear order. In general we prove that each Ramsey finite (not necessarily connected) ordered graph H has a pseudoforest as a Ramsey graph and therefore is a star forest with strong restrictions on the positions of the centers of the stars. In the asymmetric case we show that (H,H') is Ramsey finite whenever H is a so-called monotone matching. Among several further results we show that there are Ramsey finite pairs of ordered stars and ordered caterpillars of arbitrary size and diameter. This is in contrast to the unordered setting where for any Ramsey finite pair (H,H') of forests either one of H or H' is a matching or both are star forests (with additional constraints). Several of our results give a relation between Ramsey finiteness and the existence of sparse ordered Ramsey graphs. Motivated by these relations we characterize all pairs of ordered graphs that have a forest as an ordered Ramsey graph and all pairs of connected ordered graphs that have a pseudoforest as a Ramsey graph.

中文翻译：

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最小有序拉姆齐图

有序图是配备了其顶点集的线性排序的图。如果一对有序图只有有限多个最小有序 Ramsey 图，则它是 Ramsey 有限的，否则是 Ramsey 无限的。这里的有序图 F 是一对 (H,H') 有序图的有序拉姆齐图，如果对于 F 的边的任何颜色为红色和蓝色的着色，存在所有边都为红色的 H 的副本或H' 的副本，所有边缘都为蓝色。如果它的适当子图都不是 (H,H') 的有序 Ramsey 图，则这种有序 Ramsey 图是最小的。如果 H=H'，则 H 本身被称为拉姆齐有限。我们证明了连通有序图是 Ramsey 有限的当且仅当它是一个星，其中心是线性顺序中的第一个或最后一个顶点。总的来说，我们证明每个拉姆齐有限（不一定连通）有序图H都有一个伪森林作为拉姆齐图，因此是一个对恒星中心位置有很强限制的星森林。在非对称情况下，我们证明 (H,H') 是 Ramsey 有限的，只要 H 是所谓的单调匹配。在几个进一步的结果中，我们表明存在任意大小和直径的有序恒星和有序毛虫的 Ramsey 有限对。这与无序设置形成对比，其中对于任何 Ramsey 有限对 (H,H') 的森林，H 或 H' 中的一个是匹配的，或者两者都是星形森林（具有附加约束）。我们的一些结果给出了拉姆齐有限性和稀疏有序拉姆齐图的存在之间的关系。