Creature forcing and large continuum: The joy of halving

Abstract

For $f,g\inω^ω$ let $c^\forall_{f,g}$ 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 $c^\exists_{f,g}$ 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 $c^\exists_{f_ε,g_ε}=c^\forall_{f_ε,g_ε}=κ_ε$ for continuum many pairwise different cardinals $κ_ε$ and suitable pairs $(f_ε,g_ε)$. For the proof we introduce a new mixed-limit creature forcing construction.