It is well known that, with finite support iterations of ccc posets, we can obtain models where 3 or more cardinals of Cichon's diagram can be separated. For example, concerning the left side of Cichon's diagarm, it is consistent that \(\aleph_1 < add(N) < cov(N) < b < non(M)=cov(M)=c\). Nevertheless, getting the additional strict inequality \(non(M) < cov(M)\) is a challenge because subposets of \(E\), the standard ccc poset that adds an eventually different real, may add dominating reals (by Pawlikowski, 1992).
We construct a model of \(\aleph_1 < add(N) < cov(N) < b < non(M) < cov(M)=c\) with the help of chains of ultrafilters that allows to preserve certain unbounded families.
This is a joint work with M. Goldstern and S. Shelah.