Ramsey Numbers of Interval 2-chromatic Ordered Graphs

TL;DR

This paper investigates the Ramsey numbers of 2-interval chromatic ordered graphs, establishing lower bounds, exact values for specific families, and linear upper bounds for certain classes, advancing understanding of ordered graph Ramsey theory.

Contribution

It provides new lower bounds, exact Ramsey numbers for two families, and linear upper bounds for classes of 2-interval chromatic ordered graphs.

Findings

Lower bounds linear in vertices for certain 2-ichromatic graphs

Exact $t$-color Ramsey numbers for two graph families

Linear upper bounds for a class of 2-ichromatic graphs

Abstract

An ordered graph $G$ is a graph together with a specified linear ordering on the vertices, and its interval chromatic number is the minimum number of independent sets consisting of consecutive vertices that are needed to partition the vertex set. The $t$-color Ramsey number $R_t(G)$ of an ordered graph $G$ is the minimum number of vertices of an ordered complete graph such that every edge-coloring from a set of $t$ colors contains a monochromatic copy of $G$ such that the copy of $G$ preserves the original ordering on $G$. An ordered graph is $k$-ichromatic if it has interval chromatic number $k$. We obtain lower bounds linear in the number of vertices for the Ramsey numbers of certain classes of 2-ichromatic ordered graphs. We also determine the exact value of the $t$-color Ramsey number for two families of 2-ichromatic ordered graphs, and we prove a linear upper bound for a class of 2-ichromatic ordered graphs.