Extremal Combinatorics and Ramsey Theory in Structured Settings

该研究项目广泛关注组合学领域，所考虑的问题以其与几个不同数学领域的联系为特征。该项目的主要目的之一是探索这些联系，并综合来自不同领域的技术，并将它们用于解决一些长期存在的开放性问题。大部分计划中的工作都与大型离散结构的性质有关，该项目中感兴趣的问题与研究可能由此类结构建模的无序系统有关，例如相互作用粒子的集合，互联网之类的大型网络或人类大脑。有两个主要研究领域，即极值组合学和拉姆齐理论。该项目将研究涉及额外结构的极值组合学问题，重点是代数、数论和拓扑结构。该项目还将解决拉姆齐理论性质的问题，这些问题还具有加性组合、代数和遍历理论的联系。最后，本项目将探索这些领域问题之间的各种联系。要研究的具体问题包括超图的拓扑狄拉克型定理，研究自由结构内的无积集，改进一些加性组合的ramsey理论结果的界，以及研究非阿贝尔群的ramsey理论性质。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。

This research project focuses broadly on the area of combinatorics, and the questions under consideration are characterized by their connections to several different fields of mathematics. One of the main aims of this project is to explore these connections, and to synthesize techniques from different areas and bring them to bear on some longstanding open problems. Much of the planned work is concerned with the properties of large discrete structures, and the questions of interest in this project are relevant to the study of disordered systems that may be modeled by such structures, such as collections of interacting particles, large networks like the internet, or the human brain.There are two primary areas under investigation, namely extremal combinatorics and Ramsey theory. The project will investigate questions in extremal combinatorics that involve additional structure, with a focus on algebraic, number-theoretic, and topological structure. The project will also address questions of a Ramsey-theoretic nature that additionally have additive-combinatorial, algebraic, and ergodic-theoretic connections. Finally, the project will pursue various connections between problems in these areas. Specific questions to be attacked include a topological Dirac-type theorem for hypergraphs, investigating product-free sets living inside free structures, improving bounds for a few Ramsey-theoretic results of an additive-combinatorial flavor, and investigating the Ramsey-theoretic properties of non-abelian groups.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.