Which forcing axioms may be destroyed through forcings which preserve $2^{\aleph_0}=\aleph_2$?

Question

Assume ZFC; and the forcing axioms are referring to $\mathrm{MA}(\aleph_1)+2^{\aleph_0}=\aleph_2$, PFA, MM, or their variants.

I wonder if any of the axioms may be destroyed by forcing. If the forcing changes the cardinality of the reals then all of them become false so let's recede a bit and require that the forcing does not do that.

Edited for an obvious mistake.

Answer 1

Adding a single Cohen real will add a Suslin tree, which will violate any and all of these forcing axioms. (We can also just add a Suslin tree with a $\sigma$-closed forcing.)

We know that $\sf PFA$ is preserved under $\omega_2$-closed forcings, whereas $\sf MM$ is preserved under $\omega_2$ directed-closed forcings. This allows us to violate $\sf MM$ while preserving $\sf PFA$.

Of course, we can force $\sf MA+2^{\aleph_0}>\aleph_2$ with the usual c.c.c. forcing (and just a more careful choice of what the continuum can be), which will preserve $\sf MA$ but will violate $\sf PFA$ and its strengthening.

Answer 2

Asaf mentions in his answer that adding a Cohen real destroys $\mathsf{MA}$. There is a stronger fact we can prove for $\mathsf{PFA}$ and $\mathsf{MM}$: if $W$ is an outer model of $V$ with the same $\omega_2$ and $\mathsf{PFA}$ holds in both $V$ and $W$, then $\mathcal P(\omega_1)^V=\mathcal P(\omega_1)^W$. In particular, adding any real or subset of $\omega_1$ whatsoever, while preserving $\aleph_2$, destroys $\mathsf{PFA}$ (in fact, it destroys $\mathsf{BPFA}$). This is proved in

MR2231126 (2007d:03076) Andrés E. Caicedo and Boban Veličković, "The bounded proper forcing axiom and well orderings of the reals", Math. Res. Lett. 13 (2006), no. 2-3, 393–408.