On triple systems with independent neighbourhoods

Z. Füredi, Oleg Pikhurko, M. Simonovits

Research output: Contribution to journalArticle

24 Citations (Scopus)

Abstract

Let ℋ be a 3-uniform hypergraph on an $n$-element vertex set $V$. The neighbourhood of $a,b\in V$ is $N(ab):= {x: abx ∈ (ℋ)} $. Such a 3-graph has independent neighbourhoods if no $N(ab)$ contains an edge of ℋ. This is equivalent to ℋ not containing a copy of double struck F sign3,2 :={abx, aby, abz, xyz}. In this paper we prove an analogue of the Andrásfai-Erdös-Sós theorem for triangle-free graphs with minimum degree exceeding $2n/5$. It is shown that any double struck F sign-free 3-graph with minimum degree exceeding $(4/9-1/125){n} {2} is bipartite, (for n > n0), i.e., the vertices of ℋ can be split into two parts so that every triple meets both parts. This is, in fact, a Turán-type result. It solves a problem of Erdös and T.Sós, and answers a question of Mubayi and Rödl that ex(n,double struck F sign3,2)=max(α}(n-α)\left({α 2}. Here the right-hand side is 4/9{n}{3}+O(n2)$. Moreover e(ℋ)=ex(n,double struck F sign3,2) is possible only if V(ℋ)$ can be partitioned into two sets $A$ and $B$ so that each triple of ℋ intersects $A$ in exactly two vertices and $B$ in one.

Original languageEnglish
Pages (from-to)795-813
Number of pages19
JournalCombinatorics Probability and Computing
Volume14
Issue number5-6
DOIs
Publication statusPublished - Nov 2005

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Triple System
Minimum Degree
Triangle-free Graph
Uniform Hypergraph
Graph in graph theory
Intersect
Analogue
Vertex of a graph
Theorem

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computational Theory and Mathematics
  • Mathematics(all)
  • Discrete Mathematics and Combinatorics
  • Statistics and Probability

Cite this

On triple systems with independent neighbourhoods. / Füredi, Z.; Pikhurko, Oleg; Simonovits, M.

In: Combinatorics Probability and Computing, Vol. 14, No. 5-6, 11.2005, p. 795-813.

Research output: Contribution to journalArticle

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