Cardinal invariants of strongly porous sets

Abstract

In this work we study cardinal invariants of the ideal of strongly porous sets on $\tothe{2}{\omega}$. We prove that $\add{SP}{} = \omega_1$, $\cof{SP}{} = \mathfrak{c}$ and that it is consistent that $\non{SP}{} < \add{\mathcal{N}}{}$, answering questions of Hru\v{s}\'ak and Zindulka. We also find a connection between strongly porous sets on $\tothe{2}{\omega}$ and the Martin number for $\sigma$-linked partial orders, and we use this connection to construct a model where all the Martin numbers for $\sigma$-$k$-linked forcings are mutually different.