High dimensional Hoffman bound and applications in extremal combinatorics

Yuval Filmus; Konstantin Golubev; Noam Lifshitz

doi:10.5802/ALCO.190

High dimensional Hoffman bound and applications in extremal combinatorics

Yuval Filmus
, Konstantin Golubev
, Noam Lifshitz

Technion-Israel Institute of Technology
Swiss Federal Institute of Technology Zurich
Moscow Center for Fundamental and Applied Mathematics
Hebrew University of Jerusalem

Research output: Contribution to journal › Article › peer-review

6 Scopus citations

Abstract

The n-th tensor power of a graph with vertex set V is the graph on the vertex set V ⁿ, where two vertices are connected by an edge if they are connected in each coordinate. One powerful method for upper-bounding the largest independent set in a graph is the Hoffman bound, which gives an upper bound on the largest independent set of a graph in terms of its eigenvalues. In this paper we introduce the problem of upper-bounding independent sets in tensor powers of hypergraphs. We show that many prominent open problems in extremal combinatorics, such as the Turán problem for graphs and hypergraphs, can be encoded as special cases of this problem. We generalize the Hoffman bound to hypergraphs, and give several applications.

Original language	English
Pages (from-to)	1005-1026
Number of pages	22
Journal	Algebraic Combinatorics
Volume	4
Issue number	6
DOIs	https://doi.org/10.5802/ALCO.190
State	Published - 2021
Externally published	Yes

Bibliographical note

Publisher Copyright:
© 2021 The authors. All right reserved.

Funding

Acknowledgements. The first author is a Taub Fellow, and is supported by the Taub Foundation and ISF grant 1337/16. The second author is supported by the SNF grant number 20002_169106 and by the ERC grant 336283.

Funders	Funder number
Taub Foundation
Seventh Framework Programme	336283
European Commission
Schweizerischer Nationalfonds zur Förderung der Wissenschaftlichen Forschung	20002_169106
Israel Science Foundation	1337/16

Keywords

Chromatic number
Extremal set theory
Hypergraph
Independence ratio

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10.5802/ALCO.190

Cite this

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abstract = "The n-th tensor power of a graph with vertex set V is the graph on the vertex set V n, where two vertices are connected by an edge if they are connected in each coordinate. One powerful method for upper-bounding the largest independent set in a graph is the Hoffman bound, which gives an upper bound on the largest independent set of a graph in terms of its eigenvalues. In this paper we introduce the problem of upper-bounding independent sets in tensor powers of hypergraphs. We show that many prominent open problems in extremal combinatorics, such as the Tur{\'a}n problem for graphs and hypergraphs, can be encoded as special cases of this problem. We generalize the Hoffman bound to hypergraphs, and give several applications.",

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High dimensional Hoffman bound and applications in extremal combinatorics

Abstract

Bibliographical note

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Keywords

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