Container method: Difference between revisions

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{{Short description|Method in combinatorics}}

{{Draft topics|mathematics}}

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The method of hypergraph containers is a powerful tool that can help characterize the typical structure and/or answer extremal questions about families of discrete objects with a prescribed set of local constraints. Such questions arise naturally in [[extremal graph theory]], [[additive combinatorics]], [[discrete geometry]], [[coding theory]], and [[Ramsey theory]]; they include some of the most classical problems in the associated fields.

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One of the foundational problems of extremal graph theory, dating to work of Mantel in 1907 and [[Turán's theorem|Turán]] from the 1940s, asks to characterize those graphs that do not contain a copy of some fixed [[forbidden subgraph problem|forbidden]] {{math|''H''}} as a subgraph. In a different ___domain, one of the motivating questions in additive combinatorics is understanding how large a set of integers can be without containing a {{math|''k''}}-term [[arithmetic progression]], with upper bounds on this size given by [[Roth's theorem on arithmetic progressions|Roth]] (<math>k=3</math>) and [[Szemerédi's theorem|Szemerédi]] (general {{math|''k''}}).

The method of containers (in graphs) was initially pioneered by Kleitman and Winston in 1980, who bounded the number of lattices.<ref>{{cite journal |last1=Kleitman |first1= Daniel |last2=Winston |first2=Kenneth |title=The asymptotic number of lattices |journal=Annals of Discrete Mathematics |date=1980 |volume=6 |pages=243–249}}</ref> and graphs without 4-cycles<ref>{{cite journal |last1=Kleitman |first1= Daniel |last2=Winston |first2=Kenneth |title=On the number of graphs without 4-cycles |journal=Discrete Mathematics |date=1982 |volume=31 |pages=167–172}}</ref>. Container-style lemmas were independently developed by multiple mathematicians in different contexts, notably including Sapozhenko, who initially used this approach in 2002-2003 to enumerate independent sets in [[regular graphs]]<ref>{{cite journal |last1=Sapozhenko |first1=Alexander |title=The Cameron-Erdos conjecture |journal=Doklady Akademii Nauk |date=2003 |volume=393 |pages=749–752}}</ref> and sum-free sets in abelian groups<ref>{{cite journal |last1=Sapozhenko |first1=Alexander |title=Asymptotics for the number of sum-free sets in Abelian groups |journal=Doklady Akademii Nauk |date=2002 |volume=383 |pages=454–458}}</ref>.

A generalization of these ideas to a hypergraph container lemma was devised independently by Saxton and Thomason<ref>{{cite journal |last1=Saxton |first1=David |last2=Thomason |first2=Andrew |title=Hypergraph containers |journal=Inventiones mathematicae |date=2015 |volume=201 |pages=925-992}}</ref> and Balogh, Morris, and Samotij<ref>{{cite journal |last1=Balogh |first1= József |last2=Morris |first2=Robert|last3=Samotij|first3=Wojciech |title=Independent sets in hypergraphs |journal=Journal of the American Mathematical Society |date=2015 |volume=28 |pages=669-709}}</ref> in 2015, inspired by a variety of previous related work.

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===Statement===

We state the version of this lemma found in a work of Balogh, Morris, Samotij, and Saxton<ref>{{cite journal |last1=Balogh |first1= József |last2=Morris |first2=Robert|last3=Samotij|first3=Wojciech |title=Independent sets in hypergraphs |journal=Journal of the American Mathematical Society |date=2015 |volume=28 |pages=669-709}}</ref>.

Let <math>\mathcal{H}</math> be a <math>k</math>-uniform hypergraph and suppose that for every <math>l \in \{1,2,\ldots, k\}</math> and some <math>b, r \in \mathbb{N}</math>, we have that <math>\Delta_l(H) \le \left( \frac{b}{|V(H)|} \right)^{l-1} \frac{|E(H)|}{r}</math>. Then, there is a collection <math>\mathcal{C} \subset \mathcal{P}(V(H))</math> and a function <math>f: \mathcal{P}(V(H)) \rightarrow \mathcal{C}</math> such that