Let $FA\_\kappa (\mathbb{P})$ be the claim that for every family $\mathscr{D}$ of dense sets in the poset $\mathbb{P}$ with $\vert \mathscr{D} \vert = \kappa $ there is a filter $G$ such that for all $D \in \mathscr{D} $, we have $G \cap D \neq \emptyset$. So basically what I am asking for is a forcing axiom for a single poset, rather than for a family of posets as one usually does.


For which couples $(\kappa, \mathbb{P})$ is the assumption $FA\_\kappa (\mathbb{P})$ consistent?



Forcing axiom for a single poset
There are several immediate things to say.


Some instances are outright provable in ZFC:


* $\newcommand\FA{\text{FA}}\FA(\omega,\newcommand{\P}{\mathbb{P}}\P)$ is a theorem for every poset $\P$.
* $\FA(\kappa,\P)$ holds whenever $\P$ is ${<}\kappa$-closed, and indeed, merely ${\prec}\kappa$-strategically closed is sufficient.
* $\FA(\kappa,\P)$ fails whenever $\P$ is splitting and $\kappa\geq 2^{|\P|}$.


Other instances are consistent, because they follow from certain forcing axioms. Indeed, most of the standard forcing axioms are simply equivalent to the FA principle individually for a wide class of forcing notions:


* Under Martin's axiom, $\FA(\kappa,\P)$ holds for every c.c.c. $\P$ and $\kappa<2^{\aleph\_0}$. And indeed, MA is equivalent to the collection of these assertions.
* Under MM, $\FA(\omega\_1,\P)$ holds for every stationary-preserving poset $\P$, and indeed, MM is equivalent to the collection these assertions.
* Under PFA, $\FA(\omega\_1,\P)$ holds for every proper forcing $\P$, and indeed, PFA is equivalent to the collection of these assertions.


The axiom $\text{MA}(\text{Cohen})$ asserts $\FA(\kappa,\text{Cohen})$ for Cohen forcing for all $\kappa$ below the continuum, and this axiom appears now and then in applications. For example, I had used this axiom in my paper, [Infinite Wordle and the Mastermind numbers](https://arxiv.org/abs/2203.06804).


The *Martin number* $\mathfrak{m}(\P)$ is the smallest $\kappa$ such that $\FA(\kappa,\P)$ fails, and this is amongst a very useful cardinal characteristic, particularly for Cohen forcing or other common forcing notions. As Will mentions in the comments, these Martin numbers interact usefully with other common cardinal characteristics.


For someone such as yourself who seems interested in looking at $\FA(\kappa,\P)$ for specific and well-known $\P$, I would recommend investigating those relations as a way to explore the broader context of your question.


