### 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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