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Let $\kappa$ be an inaccessible cardinal. Recall that $\kappa$ is weakly compact if every tree of height $\kappa$ has either a level of size $\kappa$ or a branch of size $\kappa$.

So if $\kappa$ is a weakly compact cardinal, one has some way of controlling the behavior of trees of size $\kappa$.

Question 1: Are there analogous large cardinal properties which allow one to control the behavior of posets which are not trees? For example, complete Boolean algebras? Distributive lattices?

Question 2: Alternatively, can the tree property already be used to control the size of more general posets, such as lattices?

What I would like is to know that if I have a nice lattice $L$ of cardinality $\leq \kappa$, then under some condition on $\kappa$, there are properties of $L$ which I can check, among which would be the condition of having no $\kappa$-sized branch, implying that $L$ is in fact of cardinality $<\kappa$.

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    $\begingroup$ In the case of distributive lattices, such large cardinal notions have been discussed here. mathoverflow.net/questions/249370/… $\endgroup$
    – Joseph Van Name
    Commented Oct 24, 2021 at 15:32
  • $\begingroup$ And there are various characterizations of the notion of a weakly compact and that of a strongly compact cardinal in terms of a large cardinal version of the Boolean prime ideal theorem. $\endgroup$
    – Joseph Van Name
    Commented Oct 24, 2021 at 15:35
  • $\begingroup$ @JosephVanName Thanks! I will have to think about how that notion fits in here. I was anticipating that one would need a stronger large cardinal property to do the sort of thing I have in mind, but there the question seems to be about weakening weak compactness. I wonder if there is anything in the literature related to that MO post... $\endgroup$
    – Tim Campion
    Commented Oct 24, 2021 at 15:35
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    $\begingroup$ See onlinelibrary.wiley.com/doi/10.1002/malq.200410033 $\endgroup$
    – Mohammad Golshani
    Commented Oct 24, 2021 at 15:43

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