Extremal Combinatorics

点击翻译按钮获取中文摘要

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.