STRONG COLORINGS OVER PARTITIONS

William Chen-Mertens, Menachem Kojman, Juris Steprāns

Research output: Contribution to journalArticlepeer-review

1 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 languageEnglish
Pages (from-to)67-90
Number of pages24
JournalBulletin of Symbolic Logic
Volume27
Issue number1
DOIs
StatePublished - 1 Mar 2021

Keywords

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

ASJC Scopus subject areas

  • Philosophy
  • Logic

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