Combinatorial properties of Hechler forcing

Brendle, J., H. Judah and S. Shelah, Combinatorial properties of Hechler forcing, Annals of Pure and Applied Logic 59 (1992) 185-199. Using a notion of rank for Hechler forcing we show: (1) assuming ω^V₁ = ω^L₁, there is no real in V[d] which is eventually different from the reals in L[d], where d is Hechler over V; (2) adding one Hechler real makes the invariants on the left-hand side of Cichoń's diagram equal ω₁ and those on the right-hand side equal 2^ω and produces a maximal almost disjoint family of subsets of ω of size ω₁; (3) there is no perfect set of random reals over V in V[r][d], where r is random over V and d Hechler over V[r], thus answering a question of the first and second authors.