Extremal problems for graphs and the integers

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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).