Topics in Combinatorics

This project lies in the area of extremal and probabilistic combinatorics, and will address topical questions of interest in this field. These include: (1) Extremal properties of graphs. Consider graphs, directed graphs or hypergraphs which are extremal with respect to some characteristic, such as containing a large substructure or decomposition into such structures. How large can natural graph parameters (such as minimum degree) of such graphs be? What can we say about the structure of such graphs. (2) Structures in random graphs. In a similar vein to (1), what probability thresholds ensure that random graphs or hypergraphs have such properties? (3) Behaviour of random processes in graphs. There are many natural random processes for graphs, directed graphs and hypergraphs whose study has extensive applications. How accurately can we describe the behaviour of such processes? These topics are fundamental questions, aspects of which have attracted the attention of leading researchers around the world.