# Making the precaliber number bigger than all Knaster numbers

Write $$\mathfrak m_k$$ for the Martin's axiom number for $$k$$-Knaster, i.e., for the smallest size of a family of dense subsets of some $$k$$-Knaster poset for which there is no generic filter. (A poset is $$k$$-Knaster if every uncountable set of conditions has an uncountable $$k$$-linked subset, i.e., any $$k$$ conditions in this subset have a common lower bound.)

Write $$\mathfrak m^c$$ for the Martin's axiom number for precaliber (i.e., as above, replacing "$$k$$-linked" by "centered": every finite subset has a lower bound).

Clearly $$\mathfrak m_k\le \mathfrak m_{k+1}\le\mathfrak m^c$$.

In a recent paper arXiv:1904.02617 we show (as a side result) that consistently $$\mathfrak m_k=\aleph_1$$ for all $$k$$ while $$\mathfrak m^c=\lambda<\mu=2^{\aleph_0}$$ for any desired regular uncountable $$\lambda<\mu$$.

Question: We suspect that this result is known, and that a known proof might even use the same (nameless?) forcing notion that we use (we call it $$P_{\text{cal},\lambda}$$). Does anybody have a reference?

Details for our forcing:

• It is well known (e.g., Barnett, Fund. Math. 141, 1992) that Cohen reals add a witness for $$\mathfrak m_k=\aleph_1$$ which is preserved under further $$k+1$$-Knaster extensions. So whenever we have a FS ccc iteration which is $$\ell$$-Knaster for all $$\ell$$ and adds Cohens (at the beginning, say), the resulting model will satisfy $$\mathfrak m_k=\aleph_1$$ for all $$k$$.
• If we additionally make sure that by bookkeeping we force with all small ($$<\lambda$$) precaliber forcings, we get $$\mathfrak m^c\ge \lambda$$.
• To ensure $$\mathfrak m^c\le \lambda$$, we initially force with $$P_{\text{cal},\lambda}$$ (defined in 5.1 of the linked paper) which adds $$Q_{\text{cal}}$$ (defined in 5.3) witnessing $$\mathfrak m^c\le \lambda$$, and furthermore this witnessing-property of $$Q_{\text{cal}}$$ is preserved in FS iterations of small precaliber extensions. (For our paper we additionally need that the iterations may also contain additional iterands which are $$(\sigma,k)$$-linked for all $$k$$.)