A Survey on Ordered Ramsey Numbers
Martin Balko
TL;DR
This survey synthesizes advances in ordered Ramsey numbers across graphs, hypergraphs, and edge-ordered variants, highlighting how vertex or edge orderings interact with classical Ramsey phenomena. It surveys general bounds, then delves into specific graph classes (bounded-degree with interval chromatic number, bounded bandwidth, and particular orderings like nested matchings, alternating paths, and monotone cycles), as well as multicolor and off-diagonal settings, exposing notable gaps and open problems. A key theme is the strong influence of order parameters on growth rates, with polynomial upper bounds for certain structured families contrasted by superpolynomial lower bounds for others, such as matchings, underscoring the often striking difference from the unordered case. The hypergraph and edge-ordered sections extend these ideas, linking monotone hyperpaths to Erdős–Szekeres-type results and establishing connections to hypergraph Ramsey numbers, while revealing substantial challenges and rich directions for future work.
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.