Combinatorial properties of Hechler forcing

by Brendle and Judah and Shelah. [BJSh:477]
Annals Pure and Applied Logic, 1992
Using a notion of rank for Hechler forcing we show: 1) assuming omega_1^V = omega_1^L, 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 Cichon's diagram equal omega_1 and those on the right-hand side equal 2^omega and produces a maximal almost disjoint family of subsets of omega of size omega_1 ; 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] .

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