On the size edge-ordered Ramsey numbers of graphs

TL;DR

This paper investigates the growth of size edge-ordered Ramsey numbers for various graph families, proving non-linear growth for sparse edge-ordered book graphs and linear or near-linear growth for three other families.

Contribution

It applies Szemerédi's regularity lemma to analyze the size edge-ordered Ramsey numbers, establishing non-linear growth for sparse graphs and linear or near-linear growth for specific families.

Findings

Edge-ordered book graphs have non-linear size Ramsey numbers.

Three families of edge-ordered graphs have linear or near-linear size Ramsey numbers.

Application of Szemerédi's regularity lemma to edge-ordered graph Ramsey theory.

Abstract

For edge-ordered graphs $G^{\prec}$ and $H^{\prec}$, the size edge-ordered Ramsey number $\hat{r}_{\text{edge}}(G^{\prec}, H^{\prec})$ is defined as the smallest integer $m$ for which there exists an edge-ordered graph $F^{\prec}$ (with underlying graph $F$) having $m$ edges, such that every $2$-coloring of the edges of $F^{\prec}$ contains a monochromatic edge-ordered subgraph isomorphic to $G^{\prec}$ or a monochromatic edge-ordered subgraph isomorphic to $H^{\prec}$. Fox and Li posed a foundational question: which families of edge-ordered graphs have linear or near-linear size edge-ordered Ramsey numbers? In this paper, we apply Szemer\'edi's regularity lemma to prove that, even for sparse graph families, specifically the well-defined class of edge-ordered book graphs, the size edge-ordered Ramsey numbers of this family exhibit non-linear growth. Furthermore, we show that three families of edge-ordered graphs exhibit linear or near-linear size edge-ordered Ramsey numbers.