  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--Szemer\'edi
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.
