CAREER: Extremal Combinatorics: Methods, Problems, and Challenges

本研究项目考虑了与szemersamedi正则性方法和Ramsey理论相关的各种问题。为了解决这些问题，PI将使用一系列组合方法，这些方法最近在相关问题上取得了实质性进展。例子包括概率方法、密度增量参数、迁移参数、分析工具和嵌入技术。本项目的第一个领域涉及szemersamedi的正则性方法。在这个领域内，该项目的主要目标之一是获得三角形移除引理及其各种扩展和变体的新边界。三角形去除引理指出，任何具有次立方个数三角形的图都可以通过去除次二次个数的边而成为无三角形图。该项目的另一个主要目标是将正则性方法进一步推广到稀疏图和其他组合结构中，并获得新的应用。具体问题包括优化获得稀疏计数引理所需的伪随机条件，证明其他组合结构（如立方体）中类似的稀疏规则性结果，以及在数论和离散几何中提供新的应用，如Green-Tao定理在素数长等差数列上的扩展。这个项目的第二个方面是估算拉姆齐数。PI将致力于证明经典（完全）图和超图拉姆齐数的新边界，并证明稀疏图拉姆齐数的线性边界。本课题研究与大型网络结构相关的组合学基本问题。大型网络的例子包括互联网、Facebook、大脑、不完美的晶体和设计好的芯片。这些网络的结构对于理解网络的功能是至关重要的。前期工作表明，本项目所研究的课题具有广泛的应用前景。此外，这项工作还导致了在数学和计算机科学的许多分支中使用的强大方法的发展。例如，之前在估计拉姆齐数方面的进展导致了概率技术的发展，这些技术对计算机科学产生了巨大的影响，比如随机算法的设计。预计对这些问题的进一步研究将带来新的方法和应用。

This research project considers a variety of problems related to Szemerédi's regularity method and Ramsey theory. In tackling these problems, the PI will use a range of combinatorial methods that have recently led to substantial progress on related problems. Examples include probabilistic methods, density increment arguments, transference arguments, analytic tools, and embedding techniques. The first area in this project concerns Szemerédi's regularity method. Within this area, one of the main goals of the project is to obtain new bounds on the triangle removal lemma and its various extensions and variants. The triangle removal lemma states that any graph with a subcubic number of triangles can be made triangle-free by removing a subquadratic number of edges. Another major goal of the project is to further push the regularity method to sparse graphs and other combinatorial structures, and to obtain new applications. Specific problems include optimizing the pseudorandomness conditions needed to obtain sparse counting lemmas, proving analogous sparse regularity results in other combinatorial structures such as cubes, and providing new applications in number theory and discrete geometry such as extensions of the Green-Tao theorem on long arithmetic progressions in the primes. The second area in this project is estimating Ramsey numbers. The PI will work on proving new bounds for classical (complete) graph and hypergraph Ramsey numbers, and to prove linear bounds for Ramsey numbers of sparse graphs.This project studies fundamental problems in combinatorics related to the structure of large networks. Examples of large networks include the Internet, Facebook, the brain, imperfect crystals, and designed chips. The structure of these networks can be critical in understanding how the networks function. Previous work has shown that the subjects under study in this project have a wide range of applications. Furthermore, this work has led to the development of powerful methods that have been used in many branches of mathematics and computer science. For example, previous progress on estimating Ramsey numbers led to the development of probabilistic techniques that have had a tremendous influence on computer science, such as in the design of randomized algorithms. It is expected that further work on these problems will lead to new methods and applications.