The Martin's axiom number $\mathfrak m$ is the least cardinal $\kappa$ for which $\text{MA}\_\kappa(\text{ccc})$ is false, i.e. the least cardinal such that there exists a ccc poset $P$ and a family $\mathcal D$ of dense subsets of $P$ with $|\mathcal D| = \kappa$ such that there no $\mathcal D$-generic filter $G \subseteq P$.


Is $\mathfrak m$ regular?


All I know is that $\mathfrak m$ cannot possibly be singular unless $\mathfrak m < \mathfrak c$.
This is because 
$$\mathfrak m \leq \mathfrak p \leq \mathfrak c$$
where $\mathfrak p$ is the pseudo-intersection number. Hence $\mathfrak m = \mathfrak c$ implies $\mathfrak m = \mathfrak p$ and $\mathfrak p$ can be shown to be regular.



Is the Martin's axiom number $\mathfrak m$ regular
Not necessarily. That $\mathfrak m$ is consistently singular is proved in 



> 
> [MR0947850 (89m:03045)](http://www.ams.org/mathscinet-getitem?mr=947850) Kunen, Kenneth. [*Where $\mathsf{MA}$ first
>  fails*](http://www.jstor.org/stable/pdf/2274515.pdf). J. Symbolic Logic **53(2)**, (1988), 429–433.
> 
> 
> 


There, Ken shows that $\mathfrak{m}$ can be singular of cofinality $\omega\_1$. 


(Both links above are behind paywalls.)


