A Probabilistic Threshold for Monochromatic Arithmetic Progressions

TL;DR

This paper establishes a probabilistic threshold for the existence of monochromatic arithmetic progressions in r-colored intervals, identifying a critical interval length where such progressions almost surely appear or do not.

Contribution

It introduces a probabilistic threshold interval length, based on  and r, for the appearance of monochromatic k-term arithmetic progressions in r-colored intervals.

Findings

Threshold interval length is approximately  \, r^{k/2}.

Almost every r-coloring contains a monochromatic progression above this length.

Almost no r-coloring contains such a progression below this length.

Abstract

We show that $\sqrt{k}\cdot r^{k/2}$ is a threshold interval length where, under mild conditions, almost every $r$-coloring of an interval of longer length contains a monochromatic $k$-term arithmetic progression, while almost no $r$-coloring of an interval of shorter length contains a monochromatic $k$-term arithmetic progression.