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不同的数学分支，从它们发展出来的理论影响了不同的领域，如数论、逻辑、概率论、几何学和理论计算机科学。图的Ramsey理论有着悠久而丰富的历史，在社会研究、基础设施、电信和生物网络等领域有着广泛的应用。这一领域的研究导致了强大的技术的发展，如Szmeredi的正则性引理和相依随机选择。然而，关于超图的拉姆齐理论却鲜为人知。例如，k一致圈的Ramsey数是未知的，而它的图版从20世纪70年代就知道了。本课题的研究重点就是研究这方面的问题。另一个令人感兴趣的话题是对具有线性Ramsey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.