set theory - Martin's Axiom and Choice principles

My main questions are:

(1) Does MA have any relation to choice principles such as AC, PIT/UL, DC, AD, etc.?
(2) In particular, does MA($\kappa$) it imply AC for collections of cardinality $\kappa$?

Somewhat related...

As we all know, Martin's Axiom generalizes BCT2 (Baire category theorem for cpt. Haus. spaces). Is there an axiom which generalizes the other version of BCT, for complete metric spaces (BCT1)?

Since it is currently unknown in ZF whether BCT2 implies BCT1 (and I assume many fine attempts have been made), might it be interesting to investigate the relationship between MA and a similarly generalized version of BCT1?

Answer

The first question itself is quite easily answerable with no.
$\newcommand{\ax}[1]{\mathsf{#1}}\newcommand{\MA}{\ax{MA}}\newcommand{\DC}{\ax{DC}}\newcommand{\AD}{\ax{AD}}\newcommand{\ZF}{\ax{ZF}}\newcommand{\BPI}{\ax{BPI}}\MA$
is a very "local" axiom. It merely suggestions that a class of partial orders whose cardinality is less than the continuum satisfy a certain property.

Let $M$ be any model of $\ax{ZFC}+\MA$, with the continuum as large as you want, and we can extend it to a model of $\ZF$, $N$, in which all the choice principle mentioned fail and $\Bbb R^M=\Bbb R^N$. In particular this tells us that $\MA$ holds in $N$.

In the other direction, $\AD$ implies $\ax{CH}$ (in the sense that every uncountable set of real numbers has size continuum), so it's vacuously true that $\MA$ holds. We may want to ask whether or not $\MA(\aleph_1)$ holds, but I'm not sure how to answer that.

To the second answer, we need to add the requirement that $\kappa<2^{\aleph_0}$ (even if the real numbers cannot be well-ordered), but even then the above shows that we may have to require the sets to be sets of real numbers. That is every family of $\kappa$ non-empty sets of real numbers have a choice function. The proof is almost obvious, using the set of finite choice functions ordered by $\supseteq$. If this poset is provably c.c.c. then $\MA(\kappa)$ will prove the existence of a choice function, but at the time I don't know how to prove this is c.c.c. (and I'm not 100% sure it's even true).

Here is a paper which may be of interest. Shannon, G. P. Provable forms of Martin's Axiom, Notre Dame J. Formal Logic 31, Number 3 (1990), 382-388.