# Strong colorings over partitions

Bulletin of Symbolic Logic 27 (1):67-90 (2021)

 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 $\unicode{x3b3} <\kappa$ 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 $f\restriction )$.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 $\operatorname {Pr}_1_p$ or $\operatorname {Pr}_0_p$, also hold for an arbitrary partition p to $\theta$ parts.The symbols $$\begin{gather*} \aleph_1\nrightarrow_p[\aleph_1]^2_{\aleph_1},\;\;\; \aleph_1\nrightarrow_p[\aleph_1\circledast \aleph_1]^2_{\aleph_1},\;\;\; \aleph_0\circledast\aleph_1\nrightarrow_p[1\circledast\aleph_1]^2_{\aleph_1}, \\ \operatorname{Pr}_1_p,\;\;\;\text{ and } \;\;\; \operatorname{Pr}_0_p \end{gather*}$$ hold for an arbitrary countable partition p under the Continuum Hypothesis and are independent over ZFC $+ \neg$ CH. Keywords No keywords specified (fix it) Categories Logic and Philosophy of Logic (categorize this paper) DOI 10.1017/bsl.2021.5 Options Edit this record Mark as duplicate Export citation  Find it on Scholar Request removal from index Revision history

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## Citations of this work BETA

On the Ideal J[Κ].Assaf Rinot - 2022 - Annals of Pure and Applied Logic 173 (2):103055.

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