Extremal Combinatorics under Degree Conditions

极值组合学是组合学的一个核心领域，在近几十年中有了令人印象深刻的发展。它处理的问题是在一类有限对象上找到函数的最大值或最小值。这些问题通常与数学的其他分支和其他科学领域有关，包括生物学、化学、计算机科学、信息和编码理论。经典的极值问题，如Sperner定理和Turan定理，考虑的是集合系统或图的大小。另一方面，像Dirac关于Hamilton环的定理和Hajnal-Szemeredi定理这样的结果，考虑了图的最小或最大度。提出的研究考虑了极值集理论中几个经典结果的度版本，包括Erdos-Ko-Rado定理和Kruskal-Katona定理。解决这些问题的一个明显的困难是不能应用极值集理论中的一个重要工具——位移法。该项目还考虑了极值（超）图论中两个众所周知的难题的度版本：团密度问题和超图图图兰问题。最后，近年来将狄拉克定理推广到超图上引起了人们的广泛关注，PI将继续他在这方面的研究。PI将用于解决这些问题的工具包括代数方法、概率方法、标志代数方法、吸收方法和正则性方法。

Extremal combinatorics is a central area of combinatorics and has experienced an impressive growth in recent few decades. It deals with the problems of finding the maximum or minimum value of a function over a class of finite objects. Such problems are often related to other branches of mathematics and other fields of science, including biology, chemistry, computer science, information and coding theory. Classical extremal problems, such as Sperner's theorem and Turan's theorem, consider the size of a set system or graph. On the other hand, results such as Dirac's theorem on Hamilton cycles and the Hajnal-Szemeredi theorem, consider the minimum or maximum degree of a graph. The proposed research considers degree versions of several classical results in extremal set theory, including the Erdos-Ko-Rado theorem and the Kruskal-Katona theorem. An obvious difficulty towards solving these problems is that the shifting method, an important tool in extremal set theory, can not be applied. The project also considers degree versions of two well-known difficult problems in extremal (hyper)graph theory: the clique density problem and hypergraph Turan problem. Lastly, extending Dirac's theorem to hypergraphs has attracted much attention in recent years, and the PI will continue his research on this topic. The tools that the PI will apply to attack these problems include the algebraic method, probabilistic method, flag algebra method, absorbing method, and regularity method.