Monochromatic Progressions in Random Colorings

TL;DR

This paper investigates the thresholds for the appearance of monochromatic k-term arithmetic progressions in random 2-colorings of initial segments of natural numbers, refining previous bounds and probabilistic thresholds.

Contribution

It establishes new probabilistic bounds for the existence of monochromatic arithmetic progressions in random colorings, improving upon prior upper bounds by Brown.

Findings

Probability approaches 1 for N^{+}(k)

Probability approaches 0 for N^{-}(k)

Refines previous bounds by Brown

Abstract

Let N^{+}(k)= 2^{k/2} k^{3/2} f(k) and N^{-}(k)= 2^{k/2} k^{1/2} g(k) where 1=o(f(k)) and g(k)=o(1). We show that the probability of a random 2-coloring of {1,2,...,N^{+}(k)} containing a monochromatic k-term arithmetic progression approaches 1, and the probability of a random 2-coloring of {1,2,...,N^{-}(k)} containing a monochromatic k-term arithmetic progression approaches 0, for large k. This improves an upper bound due to Brown, who had established an analogous result for N^{+}(k)= 2^k log k f(k).