Random Van der Waerden Theorem

In this paper, we prove a sparse random analogue of the Van der Waerden Theorem. We show that, for all $r > 2$ and $q_1 \geq q_2 \dotsb q_r 3 \in \mathbb{N}$, $n^{-\frac{q_2}{q_1(q_2-1)}}$ is threshold following property: For every $r$-coloring $p$-random subset $\{1,\dotsc,n\}$, there exists monochromatic $q_i$-term arithmetic progression colored $i$, some $i$. This extends results Rödl Ruciński symmetric case = q_r$. The proof 1-statement based on Hypergraph Container Method by Balogh, Morris Samotij Saxton Thomason. 0-statement an extension Ruciński's argument case.