A famous result of Rado characterizes those integer matrices A which are partition regular, that is, for which any finite coloring of the positive integers gives rise to a monochromatic solution to the equation . Aigner-Horev and Person recently stated a conjecture on the probability threshold for the binomial random set having the asymmetric random Rado property: given partition regular matrices (for a fixed ), however one r-colors , there is always a color such that there is an i-colored solution to . This generalizes the symmetric case, which was resolved by Rödl and Ruciński, and Friedgut, Rödl and Schacht. Aigner-Horev and Person proved the 1-statement of their asymmetric conjecture. In this paper, we resolve the 0-statement in the case where the correspond to single linear equations. Additionally we close a gap in the original proof of the 0-statement of the (symmetric) random Rado theorem.

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单个方程的非对称随机 Rado 定理：0 语句

Rado 的一个著名结果表征了那些是分区规则的整数矩阵A，也就是说，对于正整数的任何有限着色都会产生方程的单色解。Aigner-Horev 和 Person 最近提出了一个关于具有非对称随机 Rado 属性的二项式随机集的概率阈值的猜想：给定分区正则矩阵（对于一个固定的），无论是一个r -colors ，总有一种颜色使得存在一个i色的解决方案. 这概括了由 Rödl 和 Ruciński 以及 Friedgut、Rödl 和 Schacht 解决的对称情况。Aigner-Horev 和 Person 证明了他们的不对称猜想的 1-陈述。在本文中，我们解决了对应于单个线性方程组的情况下的 0 语句。此外，我们在（对称）随机 Rado 定理的 0 陈述的原始证明中填补了空白。