Extremal Questions for Hypergraphs

PI将发展超图理论，或有限集族，专注于局部和全局性质之间的关系。关于大型结构中的局部/全局关系的问题影响了数学的几个领域（数论、组合学、逻辑学）以及其他领域，如信息论、编码论、理论计算机科学和社会科学。近年来，由于许多大型现实世界网络已经出现并正在研究，对这些对象的研究变得尤为重要。开发研究这些复杂系统的新技术将是未来研究人员的主要任务，而PI计划通过他的理论工作对此做出贡献。PI将集中在两个特定的领域：拉姆齐理论和超图的极值问题。在拉姆齐理论的框架内，他计划研究Erdos、Hajnal和Rado在20世纪50年代提出的关于经典超图拉姆齐数的塔形增长率的基本问题。本文还将探讨Erdos-Gyarfas-Shelah、Erdos-Rogers和Erdos-Hajnal提出的其他几个相关问题。他在极值超图理论方面的计划项目包括：解决了Kalai关于超图树极值数的一个老猜想，该猜想推广了著名的Erdos-Sos图猜想；研究凸几何问题与抽象极值超图问题的关系提出了改进Kostochka在向日葵问题上的长期界的方法；改进了已知的线性超图中环的过饱和结果，应用于Bourgain和Katz-Tao研究的加性数论中的一个问题；研究Razborov和Lovasz-Szegedy以各种形式提出的问题，即是否可以使用源于Cauchy-Schwarz不等式的方法来解决极值组合中的一大类问题。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

The PI will develop the theory of hypergraphs, or families of finite sets, focusing on the relationship between local and global properties. Questions about the local/global relationships in large structures impact several areas of mathematics (number theory, combinatorics, logic) as well as other fields like information theory, coding theory, theoretical computer science, and the social sciences. The study of these objects has gained particular importance in recent years due to the many large real world networks that have emerged and are being studied. Developing new techniques to study these complex systems will be a major task for future researchers and the PI plans to contribute to this through his theoretical work.The PI will focus on two particular areas: Ramsey theory and Extremal problems for hypergraphs. Within Ramsey theory, he plans to work on fundamental problems in the area posed by Erdos, Hajnal, and Rado starting the 1950's about the tower growth rate of classical hypergraph Ramsey numbers. Several other related problems posed by Erdos-Gyarfas-Shelah, Erdos-Rogers, and Erdos-Hajnal will also be explored. His planned projects in extremal hypergraph theory include the following: solving an old conjecture of Kalai on the extremal number of hypergraph trees that generalizes the well-known Erdos-Sos conjecture for graphs; studying the relationship between problems in convex geometry and abstract extremal hypergraph problems; developing an approach towards improving the longstanding bound of Kostochka on the sunflower problem; improving the known supersaturation results for cycles in linear hypergraphs, which has applications to a problem in additive number theory studied by Bourgain and Katz-Tao; and studying a question posed in various forms by Razborov and Lovasz-Szegedy about whether a large class of problems in extremal combinatorics can be solved using methods stemming from the Cauchy-Schwarz inequality.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.