Odd-Sunflowers

Extending the notion of sunflowers, we call a family of at least two sets an odd-sunflower if every element of the underlying set is contained in an odd number of sets or in none of them. It follows from the Erd\H os–Szemerédi conjecture, recently proved by Naslund and Sawin, that there is a constant $\mu <2$ such that every family of subsets of an $n$-element set that contains no odd-sunflower consists of at most ${\mu }^n$ sets. We construct such families of size at least ${1.5021}^n$. We also characterize minimal odd-sunflowers of triples.