A sparse canonical van der Waerden theorem

TL;DR

This work determines the threshold for the sparse canonical van der Waerden property in the binomial random set $[n]_p$, showing a sharp transition at $p \asymp n^{-1/(k-1)}$ for fixed $k\ge 3$. The authors leverage hypergraph containers, rainbow hypergraphs encoding rainbow $k$-APs, rainbow supersaturation, and sparse Szemerédi-type transfer to prove the canonical Ramsey threshold, aligning it with the standard $r$-colour vdW threshold. As a corollary, they construct sparse sets $A\subseteq [n]$ whose $k$-AP hypergraph has arbitrarily large girth while every colouring of $A$ yields a monochromatic or rainbow $k$-AP, connecting sparse Ramsey phenomena to cycle-free hypergraph structures. These results advance random Ramsey theory in additive combinatorics, providing sharp thresholds and enabling sparse canonical Ramsey constructions via probabilistic methods.

Abstract

The canonical van der Waerden theorem asserts that, for sufficiently large $n$, every colouring of $[n]$ contains either a monochromatic or a rainbow arithmetic progression of length $k$ ($k$-AP, for short). In this paper, we determine the threshold at which the binomial random subset $[n]_p$ almost surely inherits this canonical Ramsey type property. As an application, we show the existence of sets $A\subseteq [n]$ such that the $k$-APs in $A$ define a $k$-uniform hypergraph of arbitrarily high girth and yet any colouring of $A$ induces a monochromatic or rainbow $k$-AP.

Theorems & Definitions (23)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.3
• Lemma 2.2: Colour merging
• proof
• Theorem 2.3: Hypergraph containers
• Theorem 2.4: Random Szemerédi
• Theorem 2.5: Supersaturated Szemerédi
• Theorem 3.1: Random canonical van der Waerden
• Proposition 3.2: Rainbow $k$-APs in bounded colourings