Extremal problems for graphs and the integers

该项目将研究现代组合方法，并使用这些方法来解决图论和组合数理论中的问题。例如，通过正则性方法，学生将研究迫使图G包含给定大小的H-平铺的条件。(给定图H和G，图G中的H-平铺是H在G中的顶点不相交的副本的集合。)特别是，学生将致力于推广关于图中几乎完美H-平铺的Komlos的一个重要定理。该项目的另一部分将涉及整数中的极值问题(例如无和集的大小和结构)。

The project will investigate modern combinatorial methods and use these to tackle questions in graph theory as well as combinatorial number theory. For example, via the regularity method, the student will investigate conditions that force a graph G to contain an H-tiling of a given size. (Given graphs H and G, an H-tiling in a graph G is a collection of vertex-disjoint copies of H in G.) In particular, the student will work on generalising an important theorem of Komlos on almost perfect H-tilings in graphs. The other part of the project will concern extremal problems in the integers (such as the size and structure of sum-free sets).