Decisive creatures and large continuum

Abstract

For f,g∈ω^ωω let c^∀_f,g be the minimal number of uniform g-splitting trees (or: Slaloms) to cover the uniform f-splitting tree, i.e., for every branch ν of the f-tree, one of the g-trees contains ν. c^∃_f,g is the dual notion: For every branch ν, one of the g-trees guesses ν(m) infinitely often.

It is consistent that c^∃_fε,gε=c^∀_fε,gε=κ_ε for ℵ₁ many pairwise different cardinals κ_ε and suitable pairs (f_ε,g_ε).

For the proof we use creatures with sufficient bigness and halving. We show that the lim-inf creature forcing satisfies fusion and pure decision. We introduce decisiveness and use it to construct a variant of the countable support iteration of such forcings, which still satisfies fusion and pure decision.