Unrolling residues to avoid progressions

TL;DR

This paper introduces an unrolling method to extend residue-based colorings of modular groups to the set [n], effectively reducing monochromatic arithmetic progressions and improving bounds for coloring problems.

Contribution

It presents a novel unrolling technique that leverages residue colorings of rames to construct colorings of [n], achieving state-of-the-art results for small r and k.

Findings

Produced the best known colorings for minimizing monochromatic k-APs.

Extended residue colorings from rames to [n] via unrolling.

Improved bounds for coloring with small r and k.

Abstract

We consider the problem of coloring $[n]={1,2,...,n}$ with $r$ colors to minimize the number of monochromatic $k$ term arithmetic progressions (or $k$-APs for short). We show how to extend colorings of $\mathbb{Z}_m$ which avoid nontrivial $k$-APs to colorings of $[n]$ by an unrolling process. In particular, by using residues to color $\mathbb{Z}_m$ we produce the best known colorings for minimizing the number of monochromatic $k$-APs for coloring with $r$ colors for several small values of $r$ and $k$.