A sharp threshold for van der Waerden's theorem in random subsets

TL;DR

This paper proves that the threshold for the van der Waerden property in random subsets of cyclic groups is sharp, using advanced combinatorial and probabilistic methods to refine previous threshold results.

Contribution

It establishes the sharpness of the van der Waerden threshold in random subsets, building on and completing prior work by R"odl and Ruciński.

Findings

Threshold for van der Waerden property is sharp

Utilizes Friedgut's criteria for sharp thresholds

Employs container method for hypergraph independent sets

Abstract

We establish sharpness for the threshold of van der Waerden's theorem in random subsets of $\mathbb{Z}/n\mathbb{Z}$. More precisely, for $k\geq 3$ and $Z\subseteq \mathbb{Z}/n\mathbb{Z}$ we say $Z$ has the van der Waerden property if any two-colouring of $Z$ yields a monochromatic arithmetic progression of length $k$. R\"odl and Ruci\'nski (1995) determined the threshold for this property for any k and we show that this threshold is sharp. The proof is based on Friedgut's criteria (1999) for sharp thresholds, and on the recently developed container method for independent sets in hypergraphs by Balogh, Morris and Samotij (2015) and by Saxton and Thomason (2015).