Some enumerative combinatorics arising from a problem on quadratic nonresidues

If A is a finite set of cardinality n ≥ 1, 2 is the set of all subsets of A, and S is a nonempty subset of 2, we say that S has the odd-intersection property if there exists a subset N of A such that the cardinality of N ∩S is odd for each S ∈ S. Let OIP (n) denote the set of all subsets of 2 with the odd-intersection property. A nonempty set S of nonempty subsets of A is an obstruction (to the odd-intersection property) if S does not have the odd-intersection property, but all nonempty proper subsets of S do have it. Let O(n) denote the set of all obstructions that are contained in 2. This paper initiates a study of the cardinality of O(n) and OIP (n). Interest in this problem arose from previous work of the author on a combinatorial characterization of the finite subsets S of the positive integers with the following property: for infinitely many prime numbers p, S is a set of quadratic nonresidues of p.