Chromatic thresholds for pairs of graphs

TL;DR

This paper determines the exact minimum-degree density thresholds for two-color Ramsey problems involving pairs of 3-chromatic graphs, classifying all such pairs based on their chromatic properties.

Contribution

It provides a complete classification of thresholds for pairs of 3-chromatic graphs in a two-color Ramsey setting, identifying five possible threshold values.

Findings

Thresholds are exactly 2/3, 5/7, 3/4, 7/9, or 4/5.

Classification depends on chromatic thresholds and embeddability into C5-type configurations.

Precisely characterizes pairs of graphs for each threshold value.

Abstract

The chromatic threshold of a graph $H$ is the minimum-degree density above which every $H$-free graph has bounded chromatic number. We study a two-color Ramsey analogue: for graphs $H_1$ and $H_2$, we ask for the minimum-degree density above which every graph that admits a red-blue edge-coloring with no red copy of $H_1$ and no blue copy of $H_2$ has bounded chromatic number. We give a complete answer when both $H_1$ and $H_2$ are 3-chromatic. The threshold takes exactly one of the five values \[ \frac23,\quad \frac57,\quad \frac34,\quad \frac79,\quad \frac45, \] and we characterize precisely which pairs $(H_1,H_2)$ give each value. The classification is determined by the ordinary chromatic thresholds of $H_1$ and $H_2$ and by their embeddability into a hierarchy of $C_5$-type Ramsey configurations.