Collaborative Research: Extremal and Ramsey Problems for Graphs and Hypergraphs

图和超图是为对象（例如大型网络）之间的关系建模的数学结构，出现在许多应用程序中。对这些结构的研究在数学、计算机科学和工程的各个分支中都有许多应用。因此，理解这些和其他相关的数学结构是很重要的。研究这些结构的技术之一是概率推理，这在现代算法的发展和鲁棒高效通信网络的设计中至关重要。pi计划研究属于极值图和超图理论的经典问题。作为该项目的一部分，他们将培训本科生和研究生。该项目的相当一部分将采用概率推理、显式构造和规则方法的应用。pi考虑的几个问题可以追溯到Paul Erdos的工作，他塑造了离散数学的这些领域。特别地，pi将研究图和超图的图兰、拉姆齐和狄拉克型问题。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Graphs and hypergraphs are mathematical structures that model relations among objects (for example, large networks) and appear in many applications. The study of these structures has numerous applications in various branches of mathematics, computer science and engineering. Consequently, understanding these and other related mathematical structures is important. One of the techniques in the study of these structures is probabilistic reasoning, which has been crucial in the development of modern algorithms and the design of robust and efficient communication networks. The PIs plan to work on classical problems that belong to extremal graph and hypergraph theory. They will train undergraduate and graduate students as part of this project. A considerable part of the project will employ probabilistic reasoning, explicit constructions, and applications of the regularity method. Several problems considered by the PIs can be traced back to the work of Paul Erdos that has shaped these areas of discrete mathematics. In particular the PIs will work on Turan, Ramsey, and Dirac-type problems for graphs and hypergraphs.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.