The Odd and Even Intersection Properties

Abstract

A non-empty family $\mathscr{S}$ 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 $S \in \mathscr{S}$. In characterizing sets of integers that are quadratic non-residues modulo infinitely many primes, Wright asked for the number of such $\mathscr{S}$, as a function of $|A|$. We give explicit formulae.