The covering number of the strong measure zero ideal can be above almost everything else

Abstract

We show that certain type of tree forcings, including Sacks forcing, increases the covering of the strong measure zero ideal SN . As a consequence, in Sacks model, such covering number is equal to the size of the continuum, which indicates that this covering number is consistently larger than any other classical cardinal invariant of the continuum. Even more, Sacks forcing can be used to force that non(SN ) < cov(SN ) < cof(SN ), which is the first consistency result where more than two cardinal invariants associated with SN are pairwise different. Another consequence is that SN ⊆ s 0 in ZFC where s 0 denotes Marczewski's ideal.

A friendly iteration forcing that the four cardinal characteristics of $\mathcal{E}$ can be pairwise different

“…Afterwards, Brendle, the author and Mejía [BCM21] obtained that (A2) M is consistent with ZFC (without large cardinals). In another recent result in this direction, the author joint with Mejía and Rivera-Madrid [CMRM21] prove the consistency of a weak version of (A2) SN , addpSN q " nonpSN q ă covpSN q ă cofpSN q.…”