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?



Variants of Martin's axiom
Todorcevic and Velickovic ([Martin's axiom and partitions](http://www.numdam.org/article/CM_1987__63_3_391_0.pdf)) proved $P$ implies $\text{MA}\_{\aleph\_1}$.


