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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.

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Not necessarily. That $\mathfrak m$ is consistently singular is proved in

MR0947850 (89m:03045) Kunen, Kenneth. Where $\mathsf{MA}$ first fails. 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.)

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