论文标题
Ramsey属性的尖锐阈值
Sharp thresholds for Ramsey properties
论文作者
论文摘要
在这项工作中，我们开发了一个统一的框架，以建立各种拉姆齐特性的尖锐阈值结果。为了实现这一目标，我们将这些特性视为辅助超图的非色性。我们的主要技术结果给出了一系列此类超图的足够条件，以确保这种非色性特性在由顶点的随机子集引起的subhypergraphs中具有明显的阈值。 此外，我们在一些感兴趣的情况下验证了这些条件。在Ramsey Theopars的经典环境中，我们表明，对于所有$ r \ ge 2 $，在$ g_ {n，p} $中成为图形$ h $ in $ r $ in $ r $ colors的属性具有尖锐的阈值，以及包括所有cliques and Cycles和Cycles的所有$ r \ ge 2 $和所有$ h $。在算术环境中，我们还建立了对应于van der Waerden定理和Schur定理的属性的阈值的清晰度。
In this work, we develop a unified framework for establishing sharp threshold results for various Ramsey properties. To achieve this, we view such properties as non-colourability of auxiliary hypergraphs. Our main technical result gives sufficient conditions on a sequence of such hypergraphs that guarantee that this non-colourability property has a sharp threshold in subhypergraphs induced by random subsets of the vertices. Furthermore, we verify these conditions in several cases of interest. In the classical setting of Ramsey theory for graphs, we show that the property of being Ramsey for a graph $H$ in $r$ colours has a sharp threshold in $G_{n,p}$, for all $r \ge 2$ and all $H$ in a class of graphs that includes all cliques and cycles. In the arithmetic setting, we establish sharpness of thresholds for the properties corresponding to van der Waerden's theorem and Schur's theorem, also in any number of colours.
