For f,g∈ωωω 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=κε for ℵ1 many pairwise different cardinals κε and suitable pairs (fε,gε).
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.
"Decisive creatures and large continuum." J. Symbolic Logic 74 (1) 73 - 104, March 2009. https://doi.org/10.2178/jsl/1231082303