A set family $\mathcal{F}$ is a sunflower if the pairwise intersection of any two sets in $\mathcal{F}$ equals the intersection of the entire family. In extremal combinatorics, there is substantial interest in understanding how large a family needs to be to guarantee the presence of a sunflower of a given size.
Most of this work is connected to the Erdos-Rado sunflower conjecture, where a bound is desired that depends only on the size of the sets, not the universe from which the elements are drawn. I've been trying to determine if any bounds exist on the number of distinct elements in sunflower-free families (for a fixed sunflower size).
For example, given a family $\mathcal{F}$ of $w$-sets, we can obviously say that the number of distinct elements in the union of every set of $\mathcal{F}$ is $\le |\mathcal{F}|w$ (as there are that many elements in the family). However, in this case, every set is pairwise disjoint, so we have a sunflower. Hence, I'm expected there is a better bound we could argue for sunflower-free sets, but I'm having difficulty determining what this bound should be.
(I'm aware of the Erdos-Szemerdi sunflower conjecture that deals with universe instead of set sizes, but that's not quite what I'm after.)
Thank you all for your time, and please let me know if there's anything I can do to clarify my question!