Recall that


$\mathfrak{p}=\min\{|F|: F$ is a subfamily of $[\omega]^{\omega}$ with the *sfip* which has no infinite pseudo-intersection $\}$.


The cardinal $\mathfrak{q}\_0$ defined as the smallest cardinality of a subset of $\mathbf{R}$ which is not a $Q$-space.


**Q1.** Is it true that $\mathit{MA}(\omega\_1)$ iff $\omega\_1<\mathfrak{p}$ ?


**Q2.** Is there a model of set theory in which $\omega\_1=\mathfrak{q}\_0$ and a non-$\mathit{MA}(\omega\_1)$ ?



Is it true that $\mathit{MA}(\omega_1)$ iff $\omega_1<\mathfrak{p}$?
Q1: No, see [Between Martin's Axiom and Souslin's Hypothesis](https://eudml.org/doc/211011) by Kunen and Tall. Note: Bell proved in [The combinatorial principle $P(\mathfrak{c})$](https://eudml.org/doc/211293) that $\mathfrak{p}>\aleph\_1$ is equivalent to $\mathsf{MA}(\aleph\_1)$ for $\sigma$-centered partial orders.


Q2: $\mathsf{MA}(\aleph\_1)$, even $\mathfrak{p}>\aleph\_1$, implies $\mathfrak{q\_0}>\aleph\_1$, see page 162 of [Internal Cohen Extensions](https://doi.org/10.1016/0003-4843(70)90009-4) by Martin and Solovay. The principle $\mathsf{S}\_\aleph$ is proved on page 154, using a $\sigma$-centered partial order.
So, every model with $\mathfrak{q}\_0=\aleph\_1$ satisfies $\neg\mathsf{MA}(\aleph\_1)$ as well, for example every model of $\mathsf{CH}$ will do.


