CAREER: Extremal Combinatorics: Methods, Problems, and Challenges

职业：极值组合学：方法、问题和挑战

• 批准号: 1352121
• 负责人: Jacob Fox
• 金额: $ 40万
• 依托单位: Massachusetts Institute of Technology
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-04-01 至 2015-10-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1352121&HistoricalAwards=false
• 关键词: CAREER; Extremal; Combinatorics; Methods; Problems

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.

该研究项目考虑了与Szemerédi的规律性方法和Ramsey理论有关的各种问题。在解决这些问题时，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重大进展。示例包括有问题的方法，密度增量参数，转移参数，分析工具和嵌入技术。该项目的第一个领域涉及Szemerédi的规律方法。在该领域内，该项目的主要目标之一是在三角拆卸引理及其各种扩展和变体方面获得新的界限。三角拆除引理指出，任何具有亚立方数的三角形的图形都可以通过删除子分数的边缘来使三角形不含三角形。该项目的另一个主要目标是将规律方法进一步推向稀疏图和其他组合结构，并获得新的应用程序。具体问题包括优化获得稀疏计数引理所需的假性条件，提供类似的稀疏规律性导致其他组合结构（例如立方体），以及在数量理论中提供新的应用程序和离散几何形状，例如在数量的长期算术中的绿色TAO理论延迟。该项目的第二个领域是估计拉姆齐数字。 PI将致力于证明经典（完整）图和超图拉姆西数字的新界限，并证明了Ramsey数量稀疏图的线性界限。这项项目研究与大型网络结构相关的组合术中的基本问题。大型网络的示例包括互联网，Facebook，大脑，不完美的晶体和设计的芯片。这些网络的结构对于理解网络的功能至关重要。先前的工作表明，该项目中研究的受试者具有广泛的应用。此外，这项工作导致了在数学和计算机科学许多分支中使用的强大方法的发展。例如，以前在估计拉姆齐数字方面的进展导致有问题的技术的发展对计算机科学产生了巨大影响，例如在随机算法的设计中。预计在这些问题上的进一步工作将导致新的方法和应用。