Equivalence of Martin’s Axiom to its restriction to powerfully ccc posets

Determine whether Martin’s Axiom for ccc posets is equivalent to Martin’s Axiom restricted to powerfully ccc posets, i.e., posets whose every finite power is ccc.

Background

The authors investigate conditions under which the countable power X^ω of a Corson compactum is ccc yet lacks a dense metrizable subspace. This leads to considering variants of Martin’s Axiom (MA), particularly MA restricted to powerfully ccc posets, which are posets whose finite powers remain ccc.

While powerfully ccc posets suffice for all known applications of MA, it remains unresolved whether MA itself is equivalent to MA restricted to powerfully ccc posets. This unknown directly impacts the classification of Corson compacta with ccc properties in their powers, as highlighted by the equivalence results proved in the paper.