A fairly quotable special, but still representative, case of our main result is that for 2 ≤ n ≤ ω, there is a natural number m (n) such that, the following holds. Assume GCH: If $\lambda are regular, there is a cofinality preserving forcing extension in which 2 λ = μ and, for all $\sigma such that η +m(n)-1) ≤ μ, ((η +m(n)-1) ) σ ) → ((κ) σ ) η (1)n . This generalizes results of [3], Section 1, and (...) the forcing is a "many cardinals" version of the forcing there. (shrink)

A strong coloring on a cardinal$\kappa $is a function$f:[\kappa ]^2\to \kappa $such that for every$A\subseteq \kappa $of full size$\kappa $, every color$\unicode{x3b3} <\kappa $is attained by$f\restriction [A]^2$. The symbol$$ \begin{align*} \kappa\nrightarrow[\kappa]^2_{\kappa} \end{align*} $$asserts the existence of a strong coloring on$\kappa $.We introduce the symbol$$ \begin{align*} \kappa\nrightarrow_p[\kappa]^2_{\kappa} \end{align*} $$which asserts the existence of a coloring$f:[\kappa ]^2\to \kappa $which isstrong over a partition$p:[\kappa ]^2\to \theta $. A coloringfis strong overpif for every$A\in [\kappa ]^{\kappa }$there is$i<\theta $so that for every color$\unicode{x3b3} <\kappa $is (...) attained by$f\restriction ([A]^2\cap p^{-1}(i))$.We prove that whenever$\kappa \nrightarrow [\kappa ]^2_{\kappa }$holds, also$\kappa \nrightarrow _p[\kappa ]^2_{\kappa }$holds for an arbitrary finite partitionp. Similarly, arbitrary finitep-s can be added to stronger symbols which hold in any model of ZFC. If$\kappa ^{\theta }=\kappa $, then$\kappa \nrightarrow _p[\kappa ]^2_{\kappa }$and stronger symbols, like$\operatorname {Pr}_1(\kappa,\kappa,\kappa,\chi )_p$or$\operatorname {Pr}_0(\kappa,\kappa,\kappa,\aleph _0)_p$, also hold for an arbitrary partitionpto$\theta $parts.The symbols$$ \begin{gather*} \aleph_1\nrightarrow_p[\aleph_1]^2_{\aleph_1},\;\;\; \aleph_1\nrightarrow_p[\aleph_1\circledast \aleph_1]^2_{\aleph_1},\;\;\; \aleph_0\circledast\aleph_1\nrightarrow_p[1\circledast\aleph_1]^2_{\aleph_1}, \\ \operatorname{Pr}_1(\aleph_1,\aleph_1,\aleph_1,\aleph_0)_p,\;\;\;\text{ and } \;\;\; \operatorname{Pr}_0(\aleph_1,\aleph_1,\aleph_1,\aleph_0)_p \end{gather*} $$hold for an arbitrary countable partitionpunder the Continuum Hypothesis and are independent over ZFC$+ \neg $CH. (shrink)