On the Maximum Size of (p, Q)—free Families

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 a; 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 T 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 Turan type problem for (p, Q)-systems: f(n,p, Q) is denned as max ∣F∣ over all (p, Q)-free families on the ground set [n] = {1, 2,…, n}. dy 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 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)).