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

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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 is 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)247-249
Number of pages3
JournalElectronic Notes in Discrete Mathematics
Publication statusPublished - 2001

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Applied Mathematics

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