The distributivity numbers of finite products of P(ω)/fin

Generalizing [ShSp], for every n < ω we construct a ZFC-model where h(n), the distributivity number of r.o.(P(ω)/fin)n, is greater than h(n+ 1). This answers an old problem of Balcar, Pelant and Simon (see [BaPeSi]). We also show that both Laver and Miller forcings collapse the continuum to h(n) for every n < ω, hence by the first result, consistently they collapse it below h(n). Introduction. For λ a cardinal let h(λ) be the least cardinal κ for which r.o.(P(ω)/fin)λ is not κ-distributive, where by (P(ω)/fin)λ we mean the (full) λ-product of P(ω)/fin in the forcing sense; so f ∈ (P(ω)/fin)λ if and only if f : λ→ P(ω)/fin \ {0}, and the ordering is coordinatewise. In [ShSp] the consistency of h(2) < h (where h = h(1)) with ZFC has been proved, which provided a (partial) answer to a question of Balcar, Pelant and Simon in [BaPeSi]. This inequality holds in a model obtained by forcing with a countable support iteration of length ω2 of Mathias forcing over a model of GCH. That h = ω2 in this model is folklore, but the proof of h(2) = ω1 is long and difficult. The two main theorems which imply this are the following: (a) Whenever some r ∈ V Pω2 ∩ [ω] (where Pω2 is the above iteration) induces a Ramsey ultrafilter on V ∩ [ω] which is a P -filter in V Pω2 then this filter is induced by some r1 ∈ V Q0 ∩ [ω] (where Q0 is the first iterand of Pω2) and hence belongs to V Q0 . (b) Whenever some r ∈ V Q0 ∩ [ω] induces a Ramsey ultrafilter on V ∩ [ω] then this filter is Rudin–Keisler equivalent to the canonical Ramsey filter induced by the first Mathias real, and this equivalence is witnessed by some element of V ∩ ω. 1991 Mathematics Subject Classification: 03E05, 03E10, 03E35. The first author is supported by the Basic Research Foundation of the Israel Academy of Sciences; publication 531. The second author is supported by the Swiss National Science Foundation.