  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. A longstanding conjecture due to
Erd\H{o}s and Szemer\'edi states that the maximum size of a family of subsets
of $[n]$ that contains no sunflower of fixed size $k>2$ is exponentially
smaller than $2^n$ as $n\rightarrow\infty$. We consider this problem for
multiple families. In particular, we obtain sharp or almost sharp bounds on the
sum and product of $k$ families of subsets of $[n]$ that together contain no
sunflower of size $k$ with one set from each family. For the sum, we prove that
the maximum is $$(k-1)2^n+1+\sum_{s=n-k+2}^{n}\binom{n}{s}$$ for all $n \ge k
\ge 3$, and for the $k=3$ case of the product, we prove that it is between
$$\left(\frac{1}{8}+o(1)\right)2^{3n}\qquad \hbox{and} \qquad
(0.13075+o(1))2^{3n}.$$
