The odd and even intersection properties (Q640444)

Summary: A non-empty family \(\mathcal S\) of subsets of a finite set \(A\) has the odd (respectively, even) intersection property if there exists non-empty \(B \subseteq A\) with \(|B \cap S|\) odd (respectively, even) for each \(S \in \mathcal S\). In characterizing sets of integers that are quadratic nonresidues modulo infinitely many primes, Wright asked for the number of such \(\mathcal S\), as a function of \(|A|\). We give explicit formulae.