The junta method for hypergraphs and the Erdős-Chvátal simplex conjecture

Nathan Keller*, Noam Lifshitz

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

5 Scopus citations

Abstract

Numerous problems in extremal hypergraph theory ask to determine the maximal size of a k-uniform hypergraph on n vertices that does not contain an ‘enlarged’ copy H+ of a fixed hypergraph H. These include well-known problems such as the Erdős-Sós ‘forbidding one intersection’ problem and the Frankl-Füredi ‘special simplex’ problem. We present a general approach to such problems, using a ‘junta approximation method’ that originates from analysis of Boolean functions. We prove that any H+-free hypergraph is essentially contained in a ‘junta’ – a hypergraph determined by a small number of vertices – that is also H+-free, which effectively reduces the extremal problem to an easier problem on juntas. Using this approach, we obtain, for all C<k<n/C, a solution of the extremal problem for a large class of H's, which includes the aforementioned problems, and solves them for a large new set of parameters. We apply our method also to the 1974 Erdős-Chvátal simplex conjecture, which asserts that for any [Formula presented], the maximal size of a k-uniform family that does not contain a d-simplex (i.e., d+1 sets with empty intersection such that any d of them intersect) is (n−1k−1). We prove the conjecture for all d and k, provided n>n0(d).

Original languageAmerican English
Article number107991
JournalAdvances in Mathematics
Volume392
DOIs
StatePublished - 3 Dec 2021

Bibliographical note

Publisher Copyright:
© 2021 Elsevier Inc.

Keywords

  • Discrete Fourier analysis
  • Erdős-Chvátal simplex conjecture
  • Erdős-Ko-Rado theorem
  • Extremal combinatorics
  • Intersection theorems
  • Junta method

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