TY - GEN

T1 - The method of hypergraph containers

AU - Balogh, József

AU - Morris, Robert

AU - Samotij, Wojciech

N1 - Funding Information:
JB is partially supported by NSF Grant DMS-1500121 and by the Langan Scholar Fund (UIUC); RM is partially supported by CNPq (Proc. 303275/2013-8), by FAPERJ (Proc. 201.598/2014), and by ERC Starting Grant 680275 MALIG; WS is partially supported by the Israel Science Foundation grant 1147/14.
Funding Information:
JB is partially supported by NSF Grant DMS-1500121 and by the Langan Scholar Fund (UIUC); RM is partially supported by CNPq (Proc. 303275/2013-8), by FAPERJ (Proc. 201.598/2014), and by ERC Starting Grant 680275 MALIG; WS is partially supported by the Israel Science Foundation grant 1147/14. MSC2010: primary 05-02; secondary 05C30, 05C35, 05C65, 05D10, 05D40. 1A graph is H -free if it does not contain a subgraph isomorphic to H .
Publisher Copyright:
© ICM 2018.All rights reserved.

PY - 2018

Y1 - 2018

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85086373232&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85086373232&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:85086373232

T3 - Proceedings of the International Congress of Mathematicians, ICM 2018

SP - 3077

EP - 3110

BT - Invited Lectures

A2 - Sirakov, Boyan

A2 - de Souza, Paulo Ney

A2 - Viana, Marcelo

PB - World Scientific Publishing Co. Pte Ltd

T2 - 2018 International Congress of Mathematicians, ICM 2018

Y2 - 1 August 2018 through 9 August 2018

ER -