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.