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

Yury Person,Ehud Friedgut,Hiêp Hàn,Mathias Schacht

doi:10.19086/da.615

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).

Highlights

One of the main research directions in extremal and probabilistic combinatorics over the last two decades has been the extension of classical results for discrete structures to the sparse random setting
Results of that form establish the threshold for the classical result in the random setting
Every p0 ! pthe random graph Gpn, p0q with parameter p0, the probability the property holds is asymptotically zero, whereas if p0 is replaced by some p1 " pthe property does hold asymptotically almost surely (a.a.s.), i.e., for a property P of graphs and probabilities p “ ppnq we have lim PGpn, pq P P “ 0, if p ! pnÑ8

Summary

Introduction

One of the main research directions in extremal and probabilistic combinatorics over the last two decades has been the extension of classical results for discrete structures to the sparse random setting. We have to insist on the setting of Z{nZ (instead of rns) since the symmetry will play a small but crucial rôle in our proof Another shortcoming is the restriction to two colours r “ 2 and we believe it would be very interesting to extend the result to arbitrary r. Among other tools our proof relies heavily on the criterion for sharp thresholds of Friedgut and its extension due to Bourgain [4] Another crucial tool is the recent container theorem for independent sets in hypergraphs due to Balogh, Morris and Samotij [1] and Thomason and Saxton [11]. Schacht and Schulenburg [13] noted that the approach undertaken here can be refined to give a shorter proof for the sharp threshold of the Ramsey property for triangles and two colours from [7] and, more generally, for arbitrary odd cycles

Locality of coarse thresholds

Outline of the proof

Lemmas

Proof of the main theorem

Proofs of the Lemmas 6 and 10

Proof of Lemma 8

Proof of Lemma 11