随机离散结构的稳定性结果
Stability results for random discrete structures
摘要：
两年前，康伦、高尔斯和沙赫特证明了一般定理，这些定理允许人们将一大类极值组合结果从确定性转移到概率性。尽管这两篇论文解决了概率组合学中同一组长期存在的开放问题，但其中使用的方法差异很大，因此在某些方面产生的结果不具有可比性。沙赫特定理可以应用于更一般的环境，并产生更强的概率估计，而康伦和高尔斯的定理也意味着一些结构陈述的随机版本，如著名的稳定性定理的鄂尔多斯和西蒙诺维奇。在本文中，我们弥合了这两个迁移定理之间的差距。基于沙赫特的方法，我们证明了一个一般定理，允许一个转移确定性的稳定性结果的概率设置，这是更普遍和更强的比康伦和Gowers获得的。然后，我们使用这个定理推导出几个新的结果，其中一个随机版本的任意图的Erdos-Simonovits稳定性定理。主要的新想法，一个完善的方法，多次曝光时，考虑子集的二项式随机集，可能是独立的利益。
英文摘要：
Two years ago, Conlon and Gowers, and Schacht proved general theorems that allow one to transfer a large class of extremal combinatorial results from the deterministic to the probabilistic setting. Even though the two papers solve the same set of long-standing open problems in probabilistic combinatorics, the methods used in them vary significantly and therefore yield results that are not comparable in certain aspects. The theorem of Schacht can be applied in a more general setting and yields stronger probability estimates, whereas the one of Conlon and Gowers also implies random versions of some structural statements such as the famous stability theorem of Erdos and Simonovits. In this paper, we bridge the gap between these two transference theorems. Building on the approach of Schacht, we prove a general theorem that allows one to transfer deterministic stability results to the probabilistic setting that is somewhat more general and stronger than the one obtained by Conlon and Gowers. We then use this theorem to derive several new results, among them a random version of the Erdos-Simonovits stability theorem for arbitrary graphs. The main new idea, a refined approach to multiple exposure when considering subsets of binomial random sets, may be of independent interest.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
