### On partial orderings having precalibre-$\aleph_1$ and fragments of Martin's axiom

by Bagaria and Shelah. [BgSh:1041]

Fundamenta Math, 2016

We define a countable antichain condition (ccc) property for
partial orderings, weaker than precalibre- aleph_1, and show
that Martin's axiom restricted to the class of partial orderings
that have the property does not imply Martin's axiom for
sigma-linked partial orderings. This yields a new solution
to an old question of the first author about the relative strength
of Martin's axiom for sigma-centered partial orderings together
with the assertion that every Aronszajn tree is special. We also
answer a question of J. Steprans and S. Watson (1988) by showing
that, by a forcing that preserves cardinals, one can destroy
the precalibre- aleph_1 property of a partial ordering while
preserving its ccc-ness.

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