Modern extremal perspectives in random graph theory

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In recent years, the interface between combinatorics, probability and their applications has quickly developed into a rich and exciting area of research, with many applications and connections to other areas, such as theoretical computer science or statistical physics. A recent trend at this interface aims to combine different extremal results in graph theory with random graphs. Three of the main directions in this trend are the following: (1) obtaining analogues of extremal results for random graphs; (2) studying extremal problems in the union of a dense graph and a random graph, and (3) investigating how robustly graphs satisfy certain properties by analysing their random subgraphs. The aim of the project is to make progress in each of these directions, analysing both standard and non-standard models of random graphs. In particular, we will consider a new direction in randomly perturbed graphs, analyse random geometric graphs, and work extensively on random subgraphs of the hypercube. These topics are deeply related with applications in computer science, especially when the networks of interest are large.