On the maximum size of (p,Q)-free families

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2 Citations (Scopus)

Abstract

Let p be a positive integer and let Q be a subset of {0, 1,..., p}. Call p sets A1,A2,...,Ap of a ground set X a (p,Q)-system if the number of sets Ai containing x is in Q for every x ∈ X. In hypergraph terminology, a (p,Q)-system is a hypergraph with p edges such that each vertex x has degree d(x) ∈ Q. A family of sets ℱ with ground set X is called (p,Q)-free if no p sets of ℱ form a (p,Q)-system on X. We address the Turán-type problem for (p,Q)-systems: f(n,p,Q) is defined as max|ℱ| over all (p,Q)-free families on the ground set [n] = {1,2,...,n}. We study the behavior of f(n,p,Q) when p and Q are fixed (allowing 2p+1 choices for Q) while n tends to infinity. The new results of this paper mostly relate to the middle zone where 2n-1 ≤ f(n,p,Q) ≤ (1 - c)2n (in this upper bound c depends only on p). This direction was initiated by Paul Erdos who asked about the behavior of f(n,4,{0,3}). In addition, we give a brief survey on results and methods (old and recent) in the low zone (where f(n,p,Q) = o(2n)) and in the high zone (where 2n - (2 - c)n < f(n,p,Q)).

Original languageEnglish
Pages (from-to)385-403
Number of pages19
JournalDiscrete Mathematics
Volume257
Issue number2-3
DOIs
Publication statusPublished - Nov 28 2002
EventKleitman and Combinatorics: A Celebration - Cambridge, MA, United States
Duration: Aug 16 1990Aug 18 1990

Keywords

  • Degree
  • Hypergraph
  • Regular

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics

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