Generalizing [ShSp], for every n < ω we construct a ZFC-model where ℌ(n), the distributivity number of r.o.${(P\left(\omega \right)/fin)}^n$, is greater than ℌ(n+1). This answers an old problem of Balcar, Pelant and Simon (see [BaPeSi]). We also show that both Laver and Miller forcings collapse the continuum to ℌ(n) for every n < ω, hence by the first result, consistently they collapse it below ℌ(n).

In this note, we show that the model obtained by finite support iteration of a sequence of generic extensions of models of ZFC of length $\omega$ is sometimes the smallest common extension of this sequence and very often it is not.