Turan Problem for Graphs and Hypergraphs

PI建议研究图和超图的图兰问题以及相关问题。这个问题在图兰1941年的开创性论文中提出，它要求给定阶的（超）图的最大尺寸，且不包含某些固定的局部构型。一些非常令人兴奋的技术和工具在很大程度上是由turan型问题激发的，最近的发展包括稳定性方法、超图正则引理、图元和标志代数。PI将研究诸如扩大已解Turan实例族、寻找稳定性的充分条件、研究超图Turan密度的跳跃、共度问题、最小度部分版本、饱和函数以及一阶逻辑性质的Turan型问题等重要方面。本课题是极值组合学中最重要的领域之一，包括非常普遍和深刻的问题，即一些局部限制如何影响全局结构。尽管许多数学家在65年多的时间里积极尝试，但主要问题仍然没有解决。他们臭名昭著的困难并没有扼杀研究。相反，它给现代组合学的许多领域带来了生机，并揭示了与其他领域的富有成效的联系，包括线性代数、编码、设计理论、有限几何和计算机科学。希望所提出的工作将导致对这一领域的更好理解。此外，这种支持将使PI能够加强他的其他活动，例如组织研讨会，开发和教授传播新研究技术和成果的课程，以及指导年轻数学家。

The PI proposes to work on the Turan problem for graphs and hypergraphs and related questions. This problem, introduced in the seminal paper of Turan in 1941, asks for the largest size of a (hyper)graph of given order that does not contain some fixed local configuration. Some very exciting techniques and tools were greatly motivated by Turan-type questions, the recent developments including the stability approach, hypergraph regularity lemmas, graphons, and flag algebras. The PI will work on such important aspects as enlarging the family of solved Turan instances, finding good sufficient conditions for stability, studying jumps in the hypergraph Turan density, the co-degree problem, the min-degree partite version, the saturation function, and Turan-type questions for first order logic properties.The proposed topic is one of the most important areas of extremal combinatorics, comprising very general and deep problems, namely how some local restrictions can influence the global structure. The main questions are still wide open in spite of more than 65 years of active attempts by numerous mathematicians. Their notorious difficulty has not stifled research. On the contrary, it brought to life many areas of modern combinatorics and revealed fruitful connections to other fields, including linear algebra, codes, design theory, finite geometries, and computer science. Hopefully, the proposed work will lead to a better understanding of this area. Also, this support will enable the PI to enhance his other activities, such as organizing workshops, developing and teaching courses that disseminate new research techniques and results, and mentoring young mathematicians.