Generalized side-conditions and Moore–Mrówka

We prove that it is consistent (even with Martin's Axiom) that there is first-countable initially ω1ω1-compact space with cardinality greater than the continuum. We also prove that it is consistent with Martin's Axiom and c=ω2c=ω2 that there is a compact space of countable tightness which is not sequential. It is known that neither statement is consistent with the Proper Forcing Axiom. We use an innovative new method of constructing proper posets with elementary submodels as side conditions introduced by Neeman.