. L O ] 4 J an 2 00 6 DECISIVE CREATURES AND LARGE CONTINUUM

be the minimal number of uniform trees with g-splitting needed to ∀ ∞-cover a uniform tree with f-splitting. c ∃ f ,g is the dual notion for the ∃ ∞-cover. Assuming CH and given ℵ 1 many (sufficiently different) pairs (f ǫ , g ǫ) and cardinals κ ǫ such that κ ℵ 0 ǫ = κ ǫ , we construct a partial order forcing that c ∃ fǫ ,gǫ = c ∀ fǫ ,gǫ = κ ǫ. For this, we introduce a countable support semiproduct of decisive creatures with big-ness and halving. This semiproduct satisfies fusion, pure decision and continuous reading of names. 1. I While there is extensive literature on separating various cardinal characteristics with forcing, much less is known about forcing different values to many cardinal characteristics simultaneously. In the paper Many simple cardinal invariants [2], Goldstern and the second author construct a partial order P that forces pairwise different values to ℵ 1 many instances of the cardinal characteristic c ∀ f,g , defined as follows: Let f, g ∈ ω ω. An (f, g)-slalom is a sequence S = (S (n)) n∈ω such that S (n) ⊆ f (n) and |S (n)| ≤ g(n). A Family S of (f, g)-slaloms is a (∀, f, g)-cover, if for all r ∈ f there is a S ∈ S such that r(n) ∈ S (n) for all but finitely many n ∈ ω. c ∀ f,g is the smallest size of a (∀, f, g)-cover. In [2], plans to investigate the dual notion were announced as well (this investigation was promised in a paper called 448a): A Family S of (f, g)-slaloms is an (∃, f, g)-cover, if for all r ∈ f there is a S ∈ S such that r(n) ∈ S (n) for infinitely many n ∈ ω. c ∃ f,g is the smallest size of an (∃, f, g)-cover. In this paper, we assume that we have a sequence of ℵ 1 many (very different) pairs (f α , g α) α∈ω 1 and cardinals (κ α) α∈ω 1 such that κ ℵ 0 α = κ α. We also assume CH. We then construct a partial order that forces c ∃ f α ,g α = c ∀ f α ,g α = κ α for all α ∈ ω 1. Since we force 2 ℵ 0 ≥ ℵ ω 1 , …