Lecture 1 | The Hypergraph Container Method

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• Jozsef Balogh (University of Illinois at Urbana-Champaign)
• Wednesday 31 July 2024, 14:30-15:30
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HTA - Hypergraphs: Theory and Applications

In this survey we describe a recently-developed technique for bounding the number (and controlling the typical structure) of finite objects with forbidden substructures. This technique exploits a subtle clustering phenomenon exhibited by the independent sets of uniform hypergraphs whose edges are sufficiently evenly distributed; more precisely, it provides a relatively small family of ‘containers’ for the independent sets, each of which contains few edges. We attempt to convey to the reader a general high-level overview of the method, focusing on a small number of illustrative applications in areas such as extremal graph theory, Ramsey theory, additive combinatorics, and discrete geometry, and avoiding technical details as much as possible.In Lecture 1, besides motivations, we will focus on the graph container method, in Lecture 2 several applications of the Hypergraph Container Lemma will be discussed, and in Lecture 3, the sketch of the proof of the  Hypergraph Container Lemma will be discussed. Certainly, some adjustment based on Lecture 1 will be done on the later lectures.

This talk is part of the Isaac Newton Institute Seminar Series series.

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