Exact solution of the hypergraph Turán problem for k-uniform linear paths

Zoltán Füredi, Tao Jiang, Robert Seiver

Research output: Contribution to journalArticle

23 Citations (Scopus)

Abstract

A k-uniform linear path of length ℓ, denoted by ℙ(k), is a family of k-sets {F1,...,Fsuch that {pipe}Fi∩ Fi+1{pipe}=1 for each i and Fi∩ Fj = ∅ whenever {pipe}i-j{pipe}>1. Given a k-uniform hypergraph H and a positive integer n, the k-uniform hypergraph Turán number of H, denoted by exk(n, H), is the maximum number of edges in a k-uniform hypergraph F on n vertices that does not contain H as a subhypergraph. With an intensive use of the delta-system method, we determine exk(n, P(k)exactly for all fixed ℓ ≥1, k≥4, and sufficiently large n. We show that (Formula presented.). The only extremal family consists of all the k-sets in [n] that meet some fixed set of t vertices. We also show that (Formula presented.), and describe the unique extremal family. Stability results on these bounds and some related results are also established.

Original languageEnglish
Pages (from-to)299-322
Number of pages24
JournalCombinatorica
Volume34
Issue number3
DOIs
Publication statusPublished - Jun 2014

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Computational Mathematics

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