Strong colorings over partitions

Abstract

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 $\kappa \nrightarrow [\kappa {]}_{\kappa }^2$ asserts the existence of a strong coloring on $\kappa$. We introduce the symbol $\kappa {\nrightarrow }_p[\kappa {]}_{\kappa }^2$ 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 $\kappa \nrightarrow [\kappa {]}_{\kappa }^2$ holds, also $\kappa {\nrightarrow }_p[\kappa {]}_{\kappa }^2$ 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 $\kappa {\nrightarrow }_p[\kappa {]}_{\kappa }^2$ 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.

Version accepted for publication in the Bulletin of Symbolic Logic

References in corpus (1)

• Proof of a Conjecture of Galvin