From discrete to continuous: Monochromatic 3-term arithmetic progressions

TL;DR

This paper proves the optimality of a known 2-coloring scheme for minimizing monochromatic 3-term arithmetic progressions in integers, by connecting discrete colorings to continuous interval colorings and using mixed integer linear programming.

Contribution

It establishes the optimality of a specific 12-block coloring for large N among anti-symmetric colorings, using a novel connection to continuous interval colorings and permutation counting.

Findings

The known 2-coloring minimizes monochromatic 3-progressions under certain restrictions.

Optimality is proven among anti-symmetric colorings with up to 12 blocks.

The method involves permutation analysis and mixed integer linear programming.

Abstract

We prove a known 2-coloring of the integers $[N] := \{1,2,3,\ldots,N\}$ minimizes the number of monochromatic arithmetic 3-progressions under certain restrictions. A monochromatic arithmetic progression is a set of equally-spaced integers that are all the same color. Previous work by Parrilo, Robertson and Saracino conjectured an optimal coloring for large $N$ that involves 12 colored blocks. Here, we prove that the conjecture is optimal among anti-symmetric colorings with 12 or fewer colored blocks. We leverage a connection to the coloring of the continuous interval $[0,1]$ used by Parrilo, Robertson, and Saracino as well as by Butler, Costello and Graham. Our proof identifies classes of colorings with permutations, then counts the permutations using mixed integer linear programming.