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 ωV1 = ωL1, 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 2ω 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. © 1992.
Brendle, J., Judah, H., & Shelah, S. (1992). Combinatorial properties of Hechler forcing. Annals of Pure and Applied Logic, 58(3), 185–199. https://doi.org/10.1016/0168-0072(92)90027-W