# Ramsey numbers of ordered graphs under graph operations

## TL;DR

This paper studies how various graph operations influence the Ramsey numbers of ordered graphs, introducing a framework that connects extremal matrix functions to ordered Ramsey problems and providing new bounds and constructions.

## Contribution

It develops a general framework linking extremal matrix functions to ordered Ramsey numbers and applies it to derive bounds for specific ordered graphs and constructions with prescribed Ramsey growth.

## Key findings

- Upper bounds on Ramsey numbers for sum-decomposable permutations and alternating orderings
- Construction of ordered matchings with Ramsey numbers growing as n^{q+o(1)} for q in (1,2)
- Framework connecting extremal matrix functions to ordered Ramsey problems

## Abstract

An ordered graph $\mathcal{G}$ is a simple graph together with a total ordering on its vertices. The (2-color) Ramsey number of $\mathcal{G}$ is the smallest integer $N$ such that every 2-coloring of the edges of the complete ordered graph on $N$ vertices has a monochromatic copy of $\mathcal{G}$ that respects the ordering. In this paper we investigate the effect of various graph operations on the Ramsey number of a given ordered graph, and detail a general framework for applying results on extremal functions of 0-1 matrices to ordered Ramsey problems. We apply this method to give upper bounds on the Ramsey number of ordered matchings arising from sum-decomposable permutations, an alternating ordering of the cycle, and an alternating ordering of the tight hyperpath. We also construct ordered matchings on $n$ vertices whose Ramsey number is $n^{q+o(1)}$ for any given exponent $q\in(1,2)$.