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

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