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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.

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    $\begingroup$ 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? $\endgroup$ – Yemon Choi Mar 25 at 23:59
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    $\begingroup$ @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 $\endgroup$ – Tim Campion Mar 26 at 0:14
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    $\begingroup$ 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 $\endgroup$ – Joseph Van Name Mar 26 at 3:24
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    $\begingroup$ 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 $\kappa$. $\endgroup$ – Monroe Eskew Mar 27 at 7:16
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    $\begingroup$ @Monroe In (finite) combinatorics, "sunflower" is the typical expression. In set theory it is the other way. $\endgroup$ – Andrés E. Caicedo Mar 27 at 16:08

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