Minimal Ordered Ramsey Graphs

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. Our results show similarities between the ordered and the unordered setting for graphs containing cycles and significant differences for forests.