Research in Extremal Combinatorics

极值组合数学在21世纪的深度和广度上都有了显著的增长，导致了各种方法的应用远远超出了它们最初的设定。这些包括在数论、设计论、概率论、群论和理论计算机科学方面的应用，而且往往是重大突破。这里研究的问题是我们理解的障碍，在这些挑战上取得的进展和为解决这些挑战而制定的方法很可能以后会被输出到其他领域。因此，该项目的一个主要目标是开发新的工具和技术，并扩展现有方法，使其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将学习极值组合学中的各种基本问题，横跨三个主要领域：极值图论、Ramsey理论和伪随机性研究。在第一个领域中，PI打算研究几个关于极数的经典问题，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将解决图Ramsey理论中的一些经典问题，例如确定对角线Ramsey数的问题，以及算术和几何Ramsey理论中的问题。最后，PI计划在发展稀疏规则方法和寻找和应用高维扩张器方面取得进一步的进展。这些领域贯穿着共同的主题，在一个领域开发的方法可能会对另一个领域产生影响。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.