Integer colorings with no rainbow $k$-term arithmetic progression

TL;DR

This paper investigates the maximum number of rainbow $k$-term arithmetic progression-free colorings of subsets of integers, showing that the full set maximizes this count and most such colorings use only $k-1$ colors.

Contribution

It establishes that for large $n$, the full set $[n]$ maximizes rainbow $k$-AP-free colorings and characterizes the asymptotic behavior of this maximum, revealing most use only $k-1$ colors.

Findings

Maximum number of rainbow $k$-AP-free colorings occurs for the full set $[n]$.

Asymptotically, the number of such colorings grows like $(k-1)^n$ times a binomial coefficient.

Almost all rainbow $k$-AP-free colorings of $[n]$ use only $k-1$ colors.

Abstract

In this paper, we study the rainbow Erd\H{o}s-Rothschild problem with respect to $k$-term arithmetic progressions. For a set of positive integers $S \subseteq [n]$, an $r$-coloring of $S$ is \emph{rainbow $k$-AP-free} if it contains no rainbow $k$-term arithmetic progression. Let $g_{r,k}(S)$ denote the number of rainbow $k$-AP-free $r$-colorings of $S$. For sufficiently large $n$ and fixed integers $r\ge k\ge 3$, we show that $g_{r,k}(S)<g_{r,k}([n])$ for any proper subset $S\subset [n]$. Further, we prove that $\lim_{n\to \infty}g_{r,k}([n])/(k-1)^n= \binom{r}{k-1}$. Our result is asymptotically best possible and implies that, almost all rainbow $k$-AP-free $r$-colorings of $[n]$ use only $k-1$ colors.