STRONG COLORINGS OVER PARTITIONS

William Chen-Mertens; Menachem Kojman; Juris Steprāns

doi:10.1017/bsl.2021.5

STRONG COLORINGS OVER PARTITIONS

William Chen-Mertens, Menachem Kojman, Juris Steprāns

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

5 Scopus citations

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
Pages (from-to)	67-90
Number of pages	24
Journal	Bulletin of Symbolic Logic
Volume	27
Issue number	1
DOIs	https://doi.org/10.1017/bsl.2021.5
State	Published - 1 Mar 2021

Keywords

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

ASJC Scopus subject areas

Philosophy
Logic

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10.1017/bsl.2021.5

Cite this

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title = "STRONG COLORINGS OVER PARTITIONS",

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.",

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author = "William Chen-Mertens and Menachem Kojman and Juris Steprāns",

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AU - Kojman, Menachem

AU - Steprāns, Juris

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N2 - 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.

AB - 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.

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JO - Bulletin of Symbolic Logic

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STRONG COLORINGS OVER PARTITIONS

Abstract

Keywords

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