Linear paths and trees in uniform hypergraphs

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2 Citations (Scopus)

Abstract

A linear path P{double-struck}ℓ(k) is a family of k-sets {F1,..., F} such that |Fi∩Fi+1|=1 and there are no other intersections. We can represent the hyperedges by intervals. With an intensive use of the delta-system method we prove that for t>0, k>3 and sufficiently large n, (n>n0(k, t)), if F is an n-vertex k-uniform family with|F|>(n-1 k-1)+(n-2 k-1)+...+(n-t k-1), then it contains a linear path of length 2 t+1. The only extremal family consists of all edges meeting a given t-set. We also determine exk(n,P2t(k)) exactly, and the Turán number of any linear tree asymptotically.

Original languageEnglish
Pages (from-to)377-382
Number of pages6
JournalElectronic Notes in Discrete Mathematics
Volume38
DOIs
Publication statusPublished - Dec 1 2011

Keywords

  • Extremal uniform hypergraphs
  • Paths
  • Trees
  • Turán numbers

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Applied Mathematics

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