Ramsey problems for monotone paths in graphs and hypergraphs

TL;DR

This paper advances the understanding of ordered Ramsey numbers for monotone paths in graphs and hypergraphs, proving conjectures, improving bounds, and extending classical theorems in Ramsey theory.

Contribution

It proves two conjectures by Mubayi and Suk, improves existing bounds, and introduces a color-monotone version of the Canonical Ramsey Theorem.

Findings

Proved two conjectures on ordered Ramsey numbers.

Improved bounds on Ramsey numbers for monotone paths.

Established a color-monotone version of the Canonical Ramsey Theorem.

Abstract

The study of ordered Ramsey numbers of monotone paths for graphs and hypergraphs has a long history, going back to the celebrated work by Erd\H{o}s and Szekeres in the early days of Ramsey theory. In this paper we obtain several results in this area, establishing two conjectures of Mubayi and Suk and improving bounds due to Balko, Cibulka, Kr\'al and Kyn\v{c}l. We also obtain a color-monotone version of the well-known Canonical Ramsey Theorem of Erd\H{o}s and Rado, which could be of independent interest.