Coloring triangles in graphs

TL;DR

This paper investigates the minimal size of graphs that enforce monochromatic triangles in any 2-coloring, focusing on specific graph classes and their associated Ramsey numbers.

Contribution

It introduces and analyzes the Ramsey number for induced monochromatic triangles, providing bounds for particular classes of graphs.

Findings

Established bounds for $R_{ind}^{ riangle}(F)$ for certain graph classes.

Demonstrated tower-type bounds exist for these Ramsey numbers.

Provided new insights into the structure of graphs with monochromatic triangle properties.

Abstract

We study quantitative aspects of the following fact: For every graph $F$, there exists a graph $G$ with the property that any $2$-coloring of the triangles of $G$ yields an induced copy of $F$, in which all triangles are monochromatic. We define the Ramsey number $R_{\text{ind}}^{\Delta}(F)$ as the smallest size of such a graph $G$. Although this fact has several proofs, all of them provide tower-type bounds. We study the number $R_{\text{ind}}^{\Delta}(F)$ for some particular classes of graphs $F$.