
TL;DR
This paper characterizes when ordered graphs are Ramsey finite, showing that connected Ramsey finite graphs are stars with specific vertex orderings, and explores the structure of minimal ordered Ramsey graphs and pairs.
Contribution
It provides a complete characterization of Ramsey finite ordered graphs and pairs, introducing new structural results and contrasting ordered and unordered cases.
Findings
Connected Ramsey finite graphs are stars with specific vertex positions.
Every Ramsey finite ordered graph has a pseudoforest as a Ramsey graph.
There exist Ramsey finite pairs of ordered stars and caterpillars of arbitrary size.
Abstract
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…
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