Ramsey-Turán理论以及图性质的鲁棒性研究

极值图论作为图论的核心领域之一，在近几十年发展迅速并涌现出一大批顶尖数学家。经典的Turán-型问题和Dirac-型问题是极值图论中两类重要研究方向，前者主要讨论一般图的图兰数，如著名的图兰定理，而后者则探讨一般图结构存在的最小度条件，如Dirac定理。Erdős和Sós在70年代末提出的Ramsey-Turán问题，则是从图的边分布出发针对经典极值问题展开的一类重要延伸，是目前该领域的前沿热点问题。同时，为了强化该领域中的经典结论，图性质的鲁棒性研究受到了广泛关注。本项目将围绕Ramsey-Turán问题，探索一般的图因子存在的最小度条件，并从以下两方面深入探讨图性质的鲁棒性：在非兼容系统下研究团因子的存在性问题；在超图系统下探索彩色哈密顿圈存在的最小度条件。我们将引入正则方法、概率方法和吸收方法等，这些问题的解决对于极值图论的理论发展和技术创新都有非常重要的意义。

Extremal graph theory is a central area of graph theory and has experienced an impressive growth in recent few decades. It deals with the problems of finding the maximum or minimum value of a function over a class of finite objects. Such problems are often related to other branches of mathematics and other fields of science, including biology, chemistry, computer science, information and coding theory..Classical Turán problems, such as Turán’s theorem, consider the size of a graph avoiding the complete graph Kr. On the other hand, results such as Dirac’s theorem and the Hajnal-Szemerédi theorem, consider the minimum degree of a graph which ensures Hamilton cycles and Kr -factors, respectively. As a variant of Turán problem, Erdős and Sós initiated the study of Ramsey-Turán problem which excludes all graphs with large independence number. The proposed research concentrates on Ramsey-Turán problems and plans to tackle many outstanding problems in the area, including classical packing problems in the setting of low independence number, which has recently attracted much attention. The tools that will be applied to attack these problems include probabilistic method, absorbing method, and regularity method..Many typical results in graph theory are of the form “under certain conditions, G has property P”. Once such a result is established, it is natural to ask how strongly does G possess P? Recently, there has been increasing interest in the study of robustness of graph properties, aiming to strengthen classical results in extremal graph theory and probabilistic combinatorics. The project also considers robustness of graph properties and discusses two frameworks widely used to illustrate various measures of robustness: (1) Incompatibility system initiated by Krivelevich, Lee and Sudakov and (2) Graph system proposed by Aharoni and Howard. However, generalizing classical results from graphs to these frameworks is usually far from straightforward-none of them has an easy proof. Modern tools, e.g., container method, the absorbing method and the regularity method, have helped to generate new results, and yet many fundamental problems in the area remain unsolved. The project also plans to tackle many outstanding problems in the area.