  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 $\gamma<\kappa$ is attained by $f\upharpoonright[A]^2$. The symbol
$\kappa\nrightarrow [\kappa]^2_\kappa$ asserts the existence of a strong
coloring on $\kappa$.
  We introduce the symbol $\kappa\nrightarrow_p[\kappa]^2_\kappa$ 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 every color
$\gamma<\kappa$ is attained by $f\upharpoonright ([A]^2\cap p^{-1}(i))$.
  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
$\mathrm{Pr}_1(\kappa,\kappa,\kappa,\chi)$ or
$\mathrm{Pr}_0(\kappa,\kappa,\kappa,\aleph_0)$, hold also for an arbitrary
partition $p$ to $\theta$ parts.
