论文标题
随机Van der Waerden定理
Random Van der Waerden Theorem
论文作者
论文摘要
在本文中，我们证明了随机的van der waerden定理：对于$ q_1 \ geq q_2 \ geq \ geq \ dotsb \ geq q_r \ geq 3 \ in \ mathbb {n} $存在$ c，c> 0 $ （q_1，\ dotsc，q_r））= \ begin {case} 1＆\ text {if} p \ geq c \ cdot n^{ - \ frac {q_2} {q_1（q_2-1（q_2-1）}} n^{ - \ frac {q_2} {q_1（q_2-1）}}}，\ end {cases} \]扩展了Rödl和Ruciński的结果，以$ q_i = q $。 1态的证明是基于Balogh，Morris和Samotij以及Saxton和Thomason的HyperGraph容器方法。 0统计的证明是Rödl和Ruciński关于对称案例的论点的扩展。
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 Rödl and Ruciński 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 Rödl and Ruciński's argument for the symmetric case.
