极值图论及现代方法

极值组合主要研究随机性与结构性二分统一的极性现象，是组合数学领域中极具活力的重要分支，Szemerédi 获Abel奖，Bourgain，Gowers和Tao获Fields奖的工作都与极值组合相关。本项目拟研究Turán型问题、Ramsey问题、与正则引理及去除引理相关的问题、极值问题中的拟随机结构、现实网路中的极值问题等极值组合中的重要问题，涉及的现代研究方法包括代数方法、概率方法、分析方法、超图正则引理和Removal引理，对这些问题的研究将积极推动组合数学的发展。申请人和项目组成员长期从事极值组合研究，深入了解极值组合中的核心问题和现代方法，并取得了一定的研究成果。希望通过本项目的研究，发现新的研究方法，取得重要研究成果，培养从事极值组合领域研究的优秀青年人才，为提高我国组合数学的国际影响力做些贡献。

Extremal combinatorics studies the dichotomy between structure and randomness. It has become an important, influential and dynamic branch of combinatorics. Quite a number of Fields medalists, Wolf, Abel prize winners or ICM speakers are from this area. The aim of this project is to investigate a number of classical, important, and challenging topics from extremal combinatorics which include Turán type problems, Ramsey type problems, Removal lemma and its applications, Quasirandomn graphs, and Extremal problems related to real-world networks. Methods applied to this project include those from combinatorics itself, for instance, hypergraph regularity lemma and hypergraph removal lemma, together with those from Algebra, Probability theory, and Analysis. The study of topics drafted above not only has its own academic value, but also has influences on the development of other branches of mathematics as well as computer science and biology. The PI and Co-PIs have been engaged to the research of extremal combinatorics for years, and have accumulated in-depth knowledge in key techniques and problems from extremal combinatorics. Members of the group have obtained some interesting results in the study of extremal combinatorics. Several of them remain the worldwide best. Through the study of problems proposed, we aim to develop new methods and make valuable and influential theoretical contributions. We also aim to bring up some domestic talented young scholars in extremal combinatorics and enhance the global influence of Chinese combinatorics community.