Bounds on Arithmetic Rainbow Ramsey Multiplicities

TL;DR

This paper investigates how to color the first n integers with three colors to maximize rainbow 3-term arithmetic progressions, providing bounds and exact results for certain cases.

Contribution

It establishes bounds and exact maximum counts for rainbow 3-term arithmetic progressions in colored integer sets and modular integers.

Findings

Lower bounds for rainbow 3-term APs in [n]

Bounds for integers modulo n

Exact maximum when n is divisible by 3

Abstract

We study a quantitative Ramsey-type problem on 3-term arithmetic progressions: how should the set of integers $[n] = \{1, 2, \dots, n\}$ be colored using 3 colors in order to maximize the number of rainbow 3-term arithmetic progressions? By "rainbow", we mean progressions whose elements are each assigned a distinct color. We determine a lower bound for this question and upper and lower bounds when $[n]$ is replaced with the integers modulo $n$, including an exact maximum when $n$ is a multiple of 3.