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 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 Erdős 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)).