On the size edge-ordered Ramsey numbers of graphs
Yanyan Song, Yaping Mao
TL;DR
This work investigates how edge ordering affects size edge-ordered Ramsey numbers. It combines Szemer\'edi's regularity lemma with combinatorial counting to show non-linear growth for edge-ordered book graphs $B_{m,n}^{\preceq}$, establishing $\hat{r}_{\text{edge}}(B_{m,n}^{\preceq}) = \Theta(m 2^m n^2)$ for large $n$ with fixed $m$. In contrast, it identifies several bipartite families with linear or near-linear bounds, proving exact linear relations for $(m K_2^{\preceq}, P_4^{\preceq})$, and exact or tight near-linear bounds for $(2K_2^{\preceq}, K_{s,t}^{\preceq})$ and $(K_{1,n}^{\preceq}, K_{s,t}^{\preceq})$, with asymptotics $\hat{r}_{\text{edge}}(K_{s,t}^{\preceq}) = \Theta\bigl(s^2 2^{s} t\bigr)$ when $s$ is fixed and $t$ is large. The results delineate which edge-ordered graph families admit linear or near-linear size Ramsey bounds and show substantial complexity even among sparse, structured graphs.
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 Szemerédi'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.