Adding a Lot of Cohen Reals by Adding a Few

A basic fact about Cohen reals is that adding λ Cohen reals cannot produce more than λ of Cohen reals. More precisely, if 〈rα|α < λ〉 are λ-Cohen generic reals over V , then in V [〈rα|α < λ〉] there is no λ -Cohen generic real over V . But if instead of dealing with one universe V we consider two, then the above may no longer be true. The purpose of this paper is to produce models V1 ⊆ V2 such that adding κ-many Cohen reals to V2 adds λ-Cohen to V1, for some κ < λ. We deal mainly with the case when V1, V2 have same cardinals and satisfy GCH. Let us state the principal results: