Random Van der Waerden Theorem

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.