## Abstract

A strong coloring on a cardinal **Κ** is a function **ƒ : [k] ^{2 }→ Κ** such that for every

** A ⊆ κ** of full size

**κ**, every color

**γ < κ**is attained by

**ƒ ⌈ [**.

*A*]^{2}The symbol

asserts the existence of a strong coloring on .We introduce the symbol

which asserts the existence of a coloring which is strong over a partition . A coloring f is strong over p if for every there is so that for every color is attained by .We prove that whenever holds, also 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 , then and stronger symbols, like or , also hold for an arbitrary partition *p* to** θ** parts.

The symbols

hold for an arbitrary countable partition p under the Continuum Hypothesis and are independent over ZFC †¬ CH.Original language | English |
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Pages (from-to) | 67-90 |

Number of pages | 24 |

Journal | Bulletin of Symbolic Logic |

Volume | 27 |

Issue number | 1 |

DOIs | |

State | Published - 1 Mar 2021 |

## Keywords

- 2020 Mathematics Subject Classification 03E02 03E17 03E35 03E50

## ASJC Scopus subject areas

- Philosophy
- Logic