Creature forcing and large continuum: the joy of halving

A. For f , g ∈ ω let c∀f ,g be the minimal number of uniform g-splitting trees needed to cover the uniform f -splitting tree, i.e., for every branch ν of the f -tree, one of the g-trees contains ν. Let c∃f ,g be the dual notion: For every branch ν, one of the g-trees guesses ν(m) infinitely often. We show that it is consistent that c∃fǫ ,gǫ = c ∀ fǫ ,gǫ = κǫ for continuum many pairwise different cardinals κǫ and suitable pairs ( fǫ , gǫ ). For the proof we introduce a new mixed-limit creature forcing construction.