STRONG COLORINGS OVER PARTITIONS

A strong coloring on a cardinal $\kappa$ is function $f:[\kappa]^2\to \kappa$ such that for every $A\subseteq of full size $\kappa$, color $\gamma<\kappa$ attained by $f\upharpoonright[A]^2$. The symbol $\kappa\nrightarrow [\kappa]^2_\kappa$ asserts the existence $\kappa$. We introduce $\kappa\nrightarrow_p[\kappa]^2_\kappa$ which over partition $p:[\kappa]^2\to\theta$. $f$ $p$ if $A\in [\kappa]^\kappa$ there $i<\theta$ so $f\upharpoonright ([A]^2\cap p^{-1}(i))$. prove whenever $\kappa\nrightarrow[\kappa]^2_\kappa$ holds, also holds an arbitrary finite $p$. Similarly, $p$-s can be added to stronger symbols hold in any model ZFC. If $\kappa^\theta=\kappa$, then and symbols, like $\mathrm{Pr}_1(\kappa,\kappa,\kappa,\chi)$ or $\mathrm{Pr}_0(\kappa,\kappa,\kappa,\aleph_0)$, $\theta$ parts.