Problems in Extremal Set Theory

PI和他的合著者将使用组合、代数和概率方法来研究集合系统的极值问题。极值集合论的基本问题是确定一个有限集在不包含某些固定的禁止构型的情况下可以拥有的子集的最大数目。许多特例都是50多年来一直悬而未决的著名问题。然而，最近就密切相关的问题开展了大量工作，国际和平倡议是这些项目的一部分。PI将集中讨论三个主要问题：关于四点上完全3-图的Turan猜想，Chvatals关于单形的猜想，以及加强Frankl-Rodl省略的交集定理。有限集系统的一般主题与不同的数学领域(组合几何、设计理论、偏序集、加法数论)有关，也与日常生活中具体应用的学术学科(编码论、信息论、优化与调度问题、计算机科学)有关。例如，现代通信如果没有代码的存在，或者没有被设计成即使面对扭曲也能忠实地传递信息的物体，现代通信就不可能存在。集合系统的特殊问题在构造各种情况下的代码时起着重要的作用。

The PI and his coauthors will work on extremal problems for set systems using combinatorial, algebraic, and probabilistic methods. The basic problem of extremal set theory is to determine the maximum number of subsets a finite set can have without containing some fixed forbidden configuration. Many special cases are famous problems that have been open for over 50 years. Nevertheless, substantial work on closely related issues has occurred recently and the PI is part of these projects. The PI will focus on three major questions:the Turan conjecture for the complete 3-graph on four points, Chvatals conjecture on simplices, and sharpening the Frankl-Rodl omitted intersection theorem.The general topic of finite set systems has connections to diverse areas of mathematics (combinatorial geometry, design theory, partially ordered sets, additive number theory), and also to academic disciplines with concrete applications in everyday life (coding theory, information theory, optimization and scheduling problems, computer science). As an example, modern communication would be impossible without the existence of codes, or objects designed to relay information faithfully even in the face of distortion.Extremal problems for set systems play an important role in constructing codes for various situations.