Adding ${\mathrm{\aleph }}_{\omega }$ many Cohen reals

Abstractly, the generic extensions after ${\mathrm{\aleph }}_{\omega }$-many Cohen reals and ${\mathrm{\aleph }}_{\omega +1}$-many Cohen reals must be different for reasons of uniform density the relevant Boolean algebras. Nevertheless this is not satisfying and it would be nice to pin the difference between the two models down to some mathematical or combinatorial principle. In this paper we provide such a principle.