Extremal problems for multicoloured graphs and locally dependent percolation

This project is concerned with extremal problems on graphs. In a first part, we investigate a new class of Turán-type problems for multigraphs and multicoloured graphs.

We shall then use results on these problems in combination with the powerful theory of hypergraph containers in order to give a complete characterisation of typical elements in (dense) hereditary properties of multigraphs, oriented graphs, directed graphs and multicoloured graphs.

This would create a unified theory for these structures, answering theoretical questions of fundamental importance, and generalise the landmark results of Alekseev-Bollobás-Thomason and others on hereditary properties of graphs.In a second part, we consider extremal problems for locally dependent random graph models.

Our goal is to obtain a locally dependent version of the seminal Harris-Kesten theorem, one of the crown jewels of percolation theory.

The main question to answer is the extent to which local dependencies can delay the global phenomenon of percolation (i.e. the emergence of an infinite component); answers to this question have been elusive for well over a decade, despite the myriad of applications of locally dependent models in percolation theory —locally dependent models are for instance an important tool for giving rigorous estimates for critical probabilities in many well-studied percolation models— and the fundamental gap in our knowledge this represents.