Combinatorial Properties of Hechler Forcing

Using a notion of rank for Hechler forcing we show: 1) assuming ω 1 = ω L 1 , 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. 1991 Mathematics subject classification. Primary 03E40, Secondary 03E15 28A05 54H05