Incidence bounds via extremal graph theory

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"Every large system, chaotic as it may be, contains a well-organized subsystem". This phenomenon is truly ubiquitous and manifests itself in different mathematical areas. One of the central problems in extremal combinatorics, which was extensively studied in the last hundred years, is to estimate how large a graph/hypergraph needs to be to guarantee the emergence of such well-organized substructures. In the first part of this talk we will give an introduction to this topic, mentioning some classical results as well as a few applications to other areas of mathematics. We will then discuss a novel combinatorial approach, based on extremal graph theory, to a central problem in discrete geometry: counting incidences, such as point-hyperplane incidences in d-dimensional space. The study of such problems was initiated in the 1990's by Chazelle and it has interesting connections to many other topics, like additive combinatorics and theoretical computer science. This part is a joint work with Milojevic and Tomon.

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