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 is
strong over a partition
$p:[\kappa ]^2\to \theta $
. A coloring
f is strong over
p if 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 partition p. Similarly, arbitrary finite p-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 partition p to
$\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 partition
p under the Continuum Hypothesis and are independent over ZFC
$+ \neg $
CH.