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

Authors:
Zoltan Furedi;Andras Gyarfas;Miklos Ruszinko
Affiliations:
Department of Mathematics, Uiversity of Illinois at Urbana-Champaign, Urbana, IL 61801, USA and Renyi Institute of Mathematics of the Hungarian Academy of Sciences, P.O. Box 127, Budapest 1364, Hu ...;Computer and Automation Research Institute, Hungarian Academy of Sciences, P.O. Box 63, Budapest 1518, Hungary;Computer and Automation Research Institute, Hungarian Academy of Sciences, P.O. Box 63, Budapest 1518, Hungary
Venue:
Discrete Mathematics - Kleitman and combinatorics: a celebration
Year:
2002

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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 Erdös 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)).