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Let P be the statement: Every ccc partial order has $\omega_1$-precaliber; i.e., every uncountable subset $X$ of a ccc partial order $P$ has an uncountable subset $Y$ such that for every finite subset $F$ of $Y$, there is a member in $P$ below every member of $F$.

Let Q be the statement: Product of ccc partial orders is ccc.

It is known that $MA(\omega_1)$ (Martin's axiom at $\omega_1$) implies $P$ and that $P$ implies $Q$. Do these implications reverse?

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  • $\begingroup$ The new tag does not belong here, so I reverted the edit. $\endgroup$ – Andrés E. Caicedo Mar 16 '19 at 12:05
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Todorcevic and Velickovic (Martin's axiom and partitions) proved $P$ implies $\text{MA}_{\aleph_1}$.

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In a recent article of Teruyuki Yorioka it is stated that it is open whether Q implies $\text{MA}_{\aleph_1}$. The reference is [A non-implication between fragments of Martin’s Axiom related to a property which comes from Aronszajn trees, Annals of Pure and Applied Logic. 161(4), p. 469-487]. I don't think the other implications are known either.

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