Research in Extremal Combinatorics

在21世纪，极值组合学在深度和广度上都有了显著的发展，导致了一些方法的应用远远超出了它们最初的设定。其中包括数论、设计理论、概率论、群论和理论计算机科学方面的应用和重大突破。这里研究的问题代表了我们理解的障碍，很可能这些挑战的进展和解决这些挑战的方法以后可以输出到其他领域。因此，该项目的一个主要目标是开发新的工具和技术，并扩展现有的方法，以便它们适用于更一般的环境。这项工作将包括培养本科生和研究生。PI将研究极值组合学中的各种基本问题，涵盖三个主要领域：极值图论、拉姆齐理论和伪随机性研究。在第一个领域，PI打算研究几个关于极值数的经典问题，包括有理指数猜想和紧性猜想，以及不同图的同态密度之间的关系问题。在第二个领域，PI将解决一些关于图拉姆齐理论的经典问题，如确定对角线拉姆齐数，以及算术和几何拉姆齐理论的问题。最后，PI打算在稀疏正则性方法的发展以及高维展开器的寻找和应用方面取得进一步的进展。共同的主题贯穿于这些领域，在一个领域开发的方法可能会对其他领域产生影响。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。

Extremal combinatorics has grown significantly in both depth and breadth in the 21st century, resulting in methods that apply well beyond their original settings. These include applications, and often significant breakthroughs, in number theory, design theory, probability, group theory, and theoretical computer science. The questions that are studied here represent obstacles to our understanding, and it is likely that progress on these challenges and the methods developed in solving them can later be exported to other areas. As such, a major goal of this project is to develop new tools and techniques and to expand on existing methods so that they apply in more general settings. This work will involve training undergraduate and graduate students.The PI will study a variety of fundamental questions in extremal combinatorics, spanning three main areas: extremal graph theory, Ramsey theory, and the study of pseudorandomness. In the first area, the PI intends to study several classical questions on extremal numbers, including the rational exponents conjecture and the compactness conjecture, as well as problems regarding the relationships between the homomorphism densities of different graphs. In the second area, the PI will tackle some of the classical problems on graph Ramsey theory, such as that of determining diagonal Ramsey numbers, as well as questions from arithmetic and geometric Ramsey theory. Finally, the PI intends to make further progress on developing the sparse regularity method and on finding and applying high-dimensional expanders. Common themes run through these areas, and the methods developed in one area are likely to have implications for the others.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.