Sunflower / $\Delta$-system lemma in a more general poset?

The sunflower lemma (or $$\Delta$$-system lemma) may be viewed as a statement about the poset $$P_\omega(\omega_1)$$, and the generalized sunflower lemma may be viewed as a statement about the poset $$P_\lambda(\kappa)$$ for $$\kappa$$ sufficiently large compared to $$\lambda$$. Is there a version which holds in more general posets $$P$$?

If the poset $$P$$ is a lattice, then what I'm looking for is a theorem that under certain conditions, a subset $$X \subseteq P$$ has a "large" subset $$Y \subseteq X$$ whose pairwise meets are constant. If $$P$$ is not a lattice, it becomes a little bit less clear, but for example, one might state a theorem that under certain conditions on a subset $$X \subseteq P$$, there is a $$p \in P$$ and a "large" $$Y \subseteq X \cap P_{\geq p}$$ such that $$Y$$ forms a strong antichain in $$P_{\geq p}$$.

The conditions on $$X$$ would presumably say that $$X$$ is "large" and its elements are "small" in some sense.

• I'm not a combinatorist or set-theorist but I do find the current existence of a "vote to close as off-topic" rather strange. Would the voter care to explain their position, perhaps giving the OP the opportunity to improve or explain their question? – Yemon Choi Mar 25 '19 at 23:59
• @YemonChoi it’s strange — I’ve also received a downvote and earlier today I also received a downvote on an old question in a similar spirit mathoverflow.net/questions/309785/… all without explanation – Tim Campion Mar 26 '19 at 0:14
• I have been recently getting a slew of unjustified and unexplained downvotes along with delete votes. mathoverflow.net/a/320749/22277 mathoverflow.net/q/321898/22277 mathoverflow.net/q/326224/22277 mathoverflow.net/q/321894/22277 mathoverflow.net/q/321504/22277 mathoverflow.net/q/326024/22277 – Joseph Van Name Mar 26 '19 at 3:24
• Two comments. First, although I have used this lemma countless times, I never heard the name “sunflower lemma,” but only “delta-system lemma.” Second, the case with finite sets is not specific to $\omega_1$, but rather for any regular uncountable $\kappa$ we get a sunflower of size $\kappa$. – Monroe Eskew Mar 27 '19 at 7:16
• @Monroe In (finite) combinatorics, "sunflower" is the typical expression. In set theory it is the other way. – Andrés E. Caicedo Mar 27 '19 at 16:08