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This is a repost of a question that went unanswered on MSE

We say that a poset $P$ has the Knaster property (or is Knaster) if every uncountable subset of $P$ contains an uncountable subset of pairwise compatible conditions.

Let $K$ denote the statement "every c.c.c. poset is Knaster" and let $P$ denote the statement "the product of $2$ c.c.c. posets is c.c.c.". Then we have $\mathsf{MA}_{\aleph_1}\implies K \implies P$.

From the comments on the MSE question I learnt that in the paper "Forcing with a coherent souslin-tree" by Todorčević it is stated that whether the first implication is reversible is an open problem. Has there been any recent progress in the time since that paper was written or is it still open? What is known about whether the second implication is an iff instead?

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  • $\begingroup$ Both implications are open so far, if you are interested in this subject you might consult T. Yoriokas's papers. In particular "A non-implication between fragments of Martin's Axiom related to a property which comes from Aronszajn trees", and its correction. $\endgroup$ – Rahman. M Nov 13 '19 at 8:31
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    $\begingroup$ @Rahman.M thanks for the reference, that's a very interesting paper I wasn't aware of! You could post that as an answer $\endgroup$ – Alessandro Codenotti Nov 13 '19 at 10:54
  • $\begingroup$ It's fine, maybe someone really answers it in future!😉 $\endgroup$ – Rahman. M Nov 13 '19 at 14:23

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