Extremal Combinatorics

PI和他的合作者将使用组合、代数、概率和拓扑方法研究集系统或超图的极值问题。极值超图理论的基本问题是所谓的图兰问题：确定一个有限集合在不包含某些固定禁形的情况下所能拥有的子集的最大数目。总的来说，这个问题是非常困难的，某些特殊情况是50多年来一直悬而未决的著名问题。然而，最近在密切相关的问题上进行了大量的工作，并将探讨这一最近进展的局限性。虽然考虑了几个不同的方面（Ramsey-Turan问题，超图中的跳跃，简单的退化问题），但共同的主题是获得erdos - simonvits - stone定理的超图类比，erdos - simonvits - stone定理是极值图论的基石。有限集系统的一般主题与数学的各个领域（组合几何、设计理论、部分有序集、可加数论）以及在日常生活中有具体应用的学科（编码理论、信息论、优化和调度问题、计算机科学）有联系。如果没有代码的存在，现代通信是不可能的，或者即使面对失真也能忠实地传递信息的对象。集合系统的极值问题在各种情况下的代码构造中起着重要作用。

The PI and his coauthors will work on extremal problems for setsystems or hypergraphs using combinatorial, algebraic,probabilistic, and topological methods. The basic problem ofextremal hypergraph theory is the so called Turan problem:determine the maximum number of subsets a finite set can havewithout containing some fixed forbidden configuration. Thisquestion, in its full generality, is very difficult, and certainspecial cases are famous problems that have been open for over 50years. Nevertheless, substantial work on closely related issueshas occurred recently, and the limits of this recent progress willbe explored. Although several different aspects are considered(Ramsey-Turan problems, jumps in hypergraphs, degenerate problemsfor simplices), the common theme is to obtain a hypergraphanalogue of the Erdos-Simonovits-Stone theorem, the cornerstone ofextremal graph theory.The general topic of finite set systems has connections to diverseareas of mathematics (combinatorial geometry, design theory,partially ordered sets, additive number theory), and also toacademic disciplines with concrete applications in everyday life(coding theory, information theory, optimization and schedulingproblems, computer science). Modern communication would beimpossible without the existence of codes, or objects designed torelay information faithfully even in the face of distortion.Extremal problems for set systems play a role in constructingcodes for various situations.