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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|doi= 10.1016/S0167-5060(08)70708-8 |isbn= 9780444860484 }}</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 |issue= 2 |pages=167–172|doi= 10.1016/0012-365X(82)90204-7 |doi-access=free }}</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> 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> and study a variety of other enumeration problems<ref>{{Citation|last=Sapozhenko|first=Alexander|title=Systems of Containers and Enumeration Problems|date=2005|url=http://dx.doi.org/10.1007/11571155_1|work=Stochastic Algorithms: Foundations and Applications|pages=1–13|place=Berlin, Heidelberg|publisher=Springer Berlin Heidelberg|isbn=978-3-540-29498-6|access-date=2022-02-13}}</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 |issue=3 |pages=925–992|doi=10.1007/s00222-014-0562-8 |arxiv=1204.6595 |bibcode=2015InMat.201..925S |s2cid=119253715 }}</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 |issue= 3 |pages=669–709|doi= 10.1090/S0894-0347-2014-00816-X |s2cid= 15244650 |doi-access=free }}</ref> in 2015, inspired by a variety of previous related work.
==Main idea and informal statement==
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==Graph container algorithm==
We first describe a method for showing strong upper bounds on the number of independent sets in a graph; this exposition adapted from a survey of Samotij<ref>{{cite journal |last1=Samotij |first1= Wojciech |title=Counting independent sets in graphs |journal=European Journal of Combinatorics |date=2015 |volume=48 |pages=5–18|doi= 10.1016/j.ejc.2015.02.005 |s2cid= 15850625 |arxiv=1412.0940 }}</ref> about the graph container method, originally employed by Kleitman-Winston and Sapozhenko.
===Notation===
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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 |issue= 3 |pages=669–709|doi= 10.1090/S0894-0347-2014-00816-X |s2cid= 15244650 |doi-access=free }}</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
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