Sub-Ramsey numbers for arithmetic progressions

TL;DR

This paper determines the minimal maximum color class size needed to avoid totally multicolored arithmetic progressions of length three in colorings of integers, providing exact values and extremal configurations for large n.

Contribution

It establishes the exact value of f(n) for large n and characterizes all extremal colorings, answering a question posed by Alon, Caro, and Tuza.

Findings

f(n) = 8n/17 + O(1) for large n

Characterization of extremal colorings avoiding multicolored 3-term APs

Complete solution to a problem posed by Alon, Caro, and Tuza

Abstract

Let the integers $1,\ldots,n$ be assigned colors. Szemer\'edi's theorem implies that if there is a dense color class then there is an arithmetic progression of length three in that color. We study the conditions on the color classes forcing totally multicolored arithmetic progressions of length 3. Let $f(n)$ be the smallest integer $k$ such that there is a coloring of $\{1, \ldots, n\}$ without totally multicolored arithmetic progressions of length three and such that each color appears on at most $k$ integers. We provide an exact value for $f(n)$ when $n$ is sufficiently large, and all extremal colorings. In particular, we show that $f(n)= 8n/17 + O(1)$. This completely answers a question of Alon, Caro and Tuza.