Sunflower Lemma - Allow Duplicates? The sunflower lemma states that if we have a family of sets $S_1, S_2, \cdots, S_m$ such that $|S_i| \leq l$ for each $i$, then $m > (p-1)^{l+1}l!$ implies that the family contains a sunflower with $p$ petals. My question is, do the $S_i$'s need to be distinct sets or can there be duplicates (i.e $S_i = S_j$ for some $i \neq j$). Would both of these sets be a part of the sunflower in such a case?

From what I read in Wikipedia, the lemma is about a collection of sets, not a family of sets. So duplicates are disallowed.
Indeed, Let $S_0=\{1,2\}$ and $S_2=\ldots =S_m=\{1,3\}$ then no sunflower with $p=3$ petals can be formed, but we can certainly make $m>(p-1)^{l+1}l!$.
