Random Van der Waerden Theorem

TL;DR

This paper establishes a probabilistic threshold for a Van der Waerden type property in random subsets, extending previous symmetric results using hypergraph container methods.

Contribution

It generalizes the Random Van der Waerden Theorem to non-symmetric cases, providing precise probability thresholds and extending prior symmetric case results.

Findings

Threshold probability for the property is proportional to n^{-rac{q_2}{q_1(q_2-1)}}

Uses hypergraph container method for the 1-statement proof

Extends R"odl and Ruciński's argument for the 0-statement

Abstract

In this paper we prove the Random Van der Waerden Theorem: For $q_1 \geq q_2 \geq \dotsb \geq q_r \geq 3 \in \mathbb{N}$ there exist $c,C >0$ such that \[ \lim_{n \to \infty} \mathbb{P}([n]_p \rightarrow (q_1,\dotsc, q_r)) = \begin{cases} 1 & \text{if } p \geq C \cdot n^{-\frac{q_2}{q_1(q_2-1)}}, 0 & \text{if } p \leq c \cdot n^{-\frac{q_2}{q_1(q_2-1)}}, \end{cases}\] extending the results of R\"odl and Ruci\'nski for the symmetric case $q_i = q$. The proof for the 1-statement is based on the Hypergraph Container Method by Balogh, Morris and Samotij and Saxton and Thomason. The proof for the 0-statement is an extension of R\"odl and Ruci\'nski's argument for the symmetric case.