The proper forcing axiom for ℵ1-sized posets, ω1-linked symmetrically proper forcing, and the size of the continuum

In this paper, we show that the Proper Forcing Axiom for forcing notions of size [Formula: see text] is consistent with the continuum being arbitrarily large. In fact, assuming [Formula: see text] holds and [Formula: see text] is a regular cardinal, we prove that there is a proper and [Formula: see text]-c.c. forcing giving rise to a model of this forcing axiom together with [Formula: see text] and which, in addition, satisfies all statements of the form [Formula: see text], where [Formula: see text] and [Formula: see text] is a [Formula: see text] formula with the property that for every ground model [Formula: see text] of [Formula: see text] with [Formula: see text] there is, in [Formula: see text], a suitably nice poset — specifically, a poset [Formula: see text] which is [Formula: see text]-linked and symmetrically proper — adding some [Formula: see text] such that [Formula: see text]. In particular, [Formula: see text] forces Moore’s Measuring principle, Baumgartner’s Axiom for [Formula: see text]-dense sets of reals, Todorčević’s Open Coloring Axiom for sets of size [Formula: see text], the Abraham–Rubin–Shelah Open Coloring Axiom, and Todorčević’s P-ideal Dichotomy for [Formula: see text]-generated ideals on [Formula: see text], among other statements. Hence, all these statements are simultaneously compatible with a large continuum. Finally, we show that a further small variation of our construction yields a model satisfying, in addition to all the earlier conclusions, Martin’s Maximum for posets of size [Formula: see text].