Bounds on Arithmetic Rainbow Ramsey Multiplicities

TL;DR

This work addresses maximizing rainbow $3$-term arithmetic progressions under $3$-colorings, focusing on the equation $x+y=2z$. It combines explicit colorings with precise counting to show $x+y=2z$ is $3$-rainbow-uncommon over both $[n]$ and $\mathbb{Z}_n$, yielding a lower bound $\text{rb}_{[n]}(x+y=2z) \ge 2/3+o(1)$ and a range $1/3-o(1) \le \text{rb}_{\mathbb{Z}_n}(x+y=2z) \le 2/3$, with equality $=2/3$ when $n$ is a multiple of $3$. An auxiliary result gives a 3-uncommon behavior for $x+y=z$ over $\mathbb{Z}_n$, with tight bounds depending on $n\bmod 5$ and an explicit coloring achieving them. The paper highlights explicit constructions and elementary number-theoretic counts to establish asymptotic bounds and exact values, and it outlines open questions about limits and extensions to broader Ramsey-type rainbow problems.

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.

Figures (1)

• Figure 1: Graph of $x + 2d < n$ shaded with each pair $(x, d)$ representing a $3$-AP that is black if $d \neq 3t$ and white if $d = 3t$.

Theorems & Definitions (15)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1
• proof
• proof
• Theorem 2.2
• Theorem 2.3
• Theorem 2.4
• Lemma 2.5
• proof