High dimensional Hoffman bound and applications in extremal combinatorics

Yuval Filmus, Konstantin Golubev, Noam Lifshitz

Research output: Contribution to journalArticlepeer-review

5 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 languageAmerican English
Pages (from-to)1005-1026
Number of pages22
JournalAlgebraic Combinatorics
Volume4
Issue number6
DOIs
StatePublished - 2021

Bibliographical note

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

Keywords

  • Chromatic number
  • Extremal set theory
  • Hypergraph
  • Independence ratio

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