A Survey on Ordered Ramsey Numbers

TL;DR

This survey reviews recent advances in ordered Ramsey numbers, highlighting their unique behaviors, connections to other combinatorial areas, and open problems that distinguish them from unordered cases.

Contribution

It summarizes recent developments, explores connections to other fields, and presents new open problems in the study of ordered Ramsey numbers.

Findings

Ordered Ramsey numbers exhibit distinct behaviors from unordered cases.

Connections to other combinatorial areas are identified.

Several new open problems are proposed.

Abstract

The ordered Ramsey number of a graph $G^<$ with a linearly ordered vertex set is the smallest positive integer $N$ such that any two-coloring of the edges of the ordered complete graph on $N$ vertices contains a monochromatic copy of $G^<$ in the given ordering. The study of the quantitative behavior of ordered Ramsey numbers is a relatively new theme in Ramsey theory full of interesting and difficult problems. In this survey paper, we summarize recent developments in the theory of ordered Ramsey numbers. We point out connections to other areas of combinatorics and some well-known conjectures. We also list several new and challenging open problems and highlight the often strikingly different behavior from the unordered case.