Extending the notion of sunflowers, we call a family of at least two sets an \emph{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 %Alweiss, Lovett, Wu, and Zhang, 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 $1.5021^n$.