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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.

Tim Campion
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    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
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    @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 https://mathoverflow.net/questions/309785/stationarity-and-fodors-lemma-for-a-nice-poset all without explanation – Tim Campion Mar 26 '19 at 00:14
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    I have been recently getting a slew of unjustified and unexplained downvotes along with delete votes. https://mathoverflow.net/a/320749/22277 https://mathoverflow.net/q/321898/22277 https://mathoverflow.net/q/326224/22277 https://mathoverflow.net/q/321894/22277 https://mathoverflow.net/q/321504/22277 https://mathoverflow.net/q/326024/22277 – Joseph Van Name Mar 26 '19 at 03:24
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    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 07:16
  • What about, “X is a large set of elements of P, and every member of X has only finitely many elements below it”? – Monroe Eskew Mar 27 '19 at 09:25
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    @MonroeEskew Interesting! I had the impression that "$\Delta$-system" was the older and more established term, but that the alternative "sunflower" was at least well-known, sort of like "recursion theory" vs. "computability theory". I think the version of the hypotheses you suggest works well for the statement about $P_\omega(\kappa)$, but for $P_\lambda(\kappa)$ one would have to modify it to say "for each element of $X$ there are less than $2^\lambda$ elements below it" or something. Maybe it would be a bit sharper to say "there are no co-antichains of size $\lambda$ below it" or something? – Tim Campion Mar 27 '19 at 15:45
  • If the hypothesis is about the cardinality of the set of elements below any element in $X$, then getting a large antichain is just a consequence of the classical sunflower lemma, with appropriate values of $\kappa$ and $\lambda$. I’m not sure how bounding the sizes of complements of antichains below leads you to a very large antichian, but it sounds interesting. – Monroe Eskew Mar 27 '19 at 16:00
  • Sorry, I mixed-up my terminology. I should simply have said "antichain" where I said "co-antichain". I think I was thinking about dualizing the condition for an antichain to be strong but that doesn't seem to be necessary. Thanks! – Tim Campion Mar 27 '19 at 16:01
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    @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

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