Graph Theory and Combinatorics

拉姆齐理论的基本格言是，完全无序是不可能的。拉姆齐型定理也植根于数学的不同分支，从它们发展出来的理论影响了数论、逻辑学、概率论、几何和理论计算机科学等各个领域。Ramsey图论在社会研究、基础设施、电信和生物网络方面有着悠久而丰富的应用历史。这一领域的研究导致了强大技术的发展，如Szemeredi的正则引理和依赖随机选择。然而，关于超图的拉姆齐理论所知甚少。例如，k-均匀环的拉姆齐数是未知的，而它的图版本自20世纪70年代以来就已经知道了。项目的重点是研究这一领域的问题。另一个感兴趣的主题将是对具有线性拉姆齐数的稀疏超图进行分类。

The underlying motto of Ramsey theory is that total disorder is impossible. Ramsey-type theorems also have roots in different branches of mathematics, and the theory developed from them has influenced diverse areas such as number theory, logic, probability theory, geometry and theoretical computer science. Ramsey theory for graphs has a long and rich history with applications in social study, infrastructure, telecommunication and biological networks. This area of research has resulted in development of powerful techniques, such as Szemeredi's regularity lemma and dependent random choice. However, much less is known for Ramsey theory for hypergraphs. For instance the Ramsey number of k-uniform cycles are not known, whereas its graph version is known since the 1970s. The focus of the project is to study question in this area. Another topic of interest will be to classify sparse hypergraphs that has linear Ramsey number.