Extremal and probabilistic combinatorics

This project lies in the area of extremal and probabilistic combinatorics, and will address topical questions of interest in this field. These may include: (1) Dirac-type problems for quasi-random hypergraphs. There is a well-developed theory of quasi-random graphs and conditions which allow large subgraphs to be embedded within these graphs. Recently a more general classification of quasirandomness in hypergraphs has been established, bringing with it a range of interesting open problems regarding generalisations of embedding results to the hypergraph case. (2) Edit distances in graphs and hypergraphs. This is a well-studied metric on graphs and hypergraphs, for which there are still important open questions regarding, for example, the maximum edit distance of a graph or hypergraph (possibly of fixed density) from a given property. (3) Embeddings acyclic oriented graphs in tournaments. Several recent results have addressed embeddings of trees in tournaments, with the general question being to find for a given tree T the smallest n such that every tournament on n vertices contains a copy of T. This question generalises naturally by replacing `trees' with `oriented graphs not containing directed cycles', but little is known about this case so far. These topics are fundamental questions, aspects of which have attracted the attention of leading researchers around the world.