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arXiv · 1801.04584

The method of hypergraph containers

József Balogh·Robert Morris·Wojciech Samotij
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Abstract

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.

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József Balogh, Robert Morris, Wojciech Samotij. 2018-01-14. The method of hypergraph containers. https://arxiv.org/abs/1801.04584

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