Integer colorings with no rainbow 3-term arithmetic progression

TL;DR

This paper investigates the number and structure of colorings of sets and groups that avoid rainbow 3-term arithmetic progressions, revealing asymptotic counts, typical colorings, and extremal subsets.

Contribution

It provides the first asymptotic enumeration of rainbow 3-term AP-free colorings, characterizes typical colorings, and determines extremal subsets and exact counts for cyclic groups.

Findings

Asymptotic number of rainbow 3-AP-free r-colorings of [n]

Typical rainbow 3-AP-free colorings are 2-colorings

Maximum number of such colorings achieved by [n]

Abstract

In this paper, we study the rainbow Erd\H{o}s-Rothschild problem with respect to 3-term arithmetic progressions. We obtain the asymptotic number of $r$-colorings of $[n]$ without rainbow 3-term arithmetic progressions, and we show that the typical colorings with this property are 2-colorings. We also prove that $[n]$ attains the maximum number of rainbow 3-term arithmetic progression-free $r$-colorings among all subsets of $[n]$. Moreover, the exact number of rainbow 3-term arithmetic progression-free $r$-colorings of $\mathbb{Z}_p$ is obtained, where $p$ is any prime and $\mathbb{Z}_p$ is the cyclic group of order $p$.