Seminar

A combinatorial structure is said to be quasirandom if it resembles a random structure in a certain robust sense. Classical work of Rödl, Thomason, Chung, Graham and Wilson from the 1980s led to the notion of quasirandom graphs, which is nowadays considered to be well-understood. In this talk, we first review classical and recent results on quasirandom combinatorial structures, and we then focus on problems from extremal combinatorics with additional quasirandom constraints. The study of such extremal problems was initiated by Erdős and Sós in the early 1980s, however, substantial progress appeared only recently with use of the hypergraph regularity method, which was independently developed by Kohayakawa, Nagle, Rödl, Schacht and Skokan, and Gowers. We will present some of recent results, e.g. a solution of a 40-year-old problem of Erdős and Sós concerning the uniform Turán densities of K_4^3-, introduce methods developed that have been developed to tackle extremal problems with quasirandom constraints, and discuss some of many open problems concerning extremal problems with quasirandom constraints. The talk will include results obtained jointly with various collaborators, particularly, with Matija Bucić, Jacob W. Cooper, Frederik Garbe, Daniel Iľkovič, Filip Kučerák, Ander Lamaison, Samuel Mohr, David Munhá Correia and Gábor Tardos.