A sunflower is a collection of distinct sets such that the intersection of any two of them is the same as the common intersection C of all of them, and |C| is smaller than each of the sets. There has been a lot of recent progress on determining the maximum size of a sunflower-free family of subsets of [n]. We consider the problems of determining the maximum sum and product of k families of subsets of [n] that contain no sunflower of size k with one set from each family. We solve the sum problem exactly and make partial progress towards the product problem. \noindent{Joint work with Lujia Wang.}