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A club guessing toolbox I

Tanmay Inamdar, Assaf Rinot

TL;DR

This paper develops a comprehensive toolbox for club-guessing at uncountable cardinals, formulating the framework $\mathcal{CG}_\xi(S,T,\sigma,\vec{J})$ to capture a wide spectrum of guessing properties. It combines postprocessing, colorings, and coherence concepts to construct and partition $C$-sequences with strong guessing against clubs, while also enabling frequency upgrades (increasing $\sigma$) and transfers between ideals (from $J^{\text{bd}}$ to $\mathrm{NS}_\delta$). Key contributions include existence results for $\lambda$-bounded $C$-sequences that guess stationarily often against clubs, coherent and $\sqsubseteq^*$-coherent variants, and robust partitioned club-guessing via coloring principles, with implications such as strengthened PCF-type bounds and applications to higher-cardinal combinatorics. The methods unify and extend prior results (Shelah's facts) and lay groundwork for Part II, offering concrete techniques for constructing combinatorial objects and analyzing guessing frequencies at large cardinals.

Abstract

Club guessing principles were introduced by Shelah as a weakening of Jensen's diamond. Most spectacularly, they were used to prove Shelah's ZFC bound on the power of the first singular cardinal. These principles have found many other applications: in cardinal arithmetic and PCF theory; in the construction of combinatorial objects on uncountable cardinals such as Jonsson algebras, strong colourings, Souslin trees, and pathological graphs; to the non-existence of universals in model theory; to the non-existence of forcing axioms at higher uncountable cardinals; and many more. In this paper, the first part of a series, we survey various forms of club-guessing that have appeared in the literature, and then systematically study the various ways in which a club-guessing sequences can be improved, especially in the way the frequency of guessing is calibrated. We include an expository section intended for those unfamiliar with club-guessing and which can be read independently of the rest of the article.

A club guessing toolbox I

TL;DR

This paper develops a comprehensive toolbox for club-guessing at uncountable cardinals, formulating the framework to capture a wide spectrum of guessing properties. It combines postprocessing, colorings, and coherence concepts to construct and partition -sequences with strong guessing against clubs, while also enabling frequency upgrades (increasing ) and transfers between ideals (from to ). Key contributions include existence results for -bounded -sequences that guess stationarily often against clubs, coherent and -coherent variants, and robust partitioned club-guessing via coloring principles, with implications such as strengthened PCF-type bounds and applications to higher-cardinal combinatorics. The methods unify and extend prior results (Shelah's facts) and lay groundwork for Part II, offering concrete techniques for constructing combinatorial objects and analyzing guessing frequencies at large cardinals.

Abstract

Club guessing principles were introduced by Shelah as a weakening of Jensen's diamond. Most spectacularly, they were used to prove Shelah's ZFC bound on the power of the first singular cardinal. These principles have found many other applications: in cardinal arithmetic and PCF theory; in the construction of combinatorial objects on uncountable cardinals such as Jonsson algebras, strong colourings, Souslin trees, and pathological graphs; to the non-existence of universals in model theory; to the non-existence of forcing axioms at higher uncountable cardinals; and many more. In this paper, the first part of a series, we survey various forms of club-guessing that have appeared in the literature, and then systematically study the various ways in which a club-guessing sequences can be improved, especially in the way the frequency of guessing is calibrated. We include an expository section intended for those unfamiliar with club-guessing and which can be read independently of the rest of the article.
Paper Structure (16 sections, 54 theorems, 123 equations)

This paper contains 16 sections, 54 theorems, 123 equations.

Table of Contents

  1. Introduction
  2. Motivation
  3. The results
  4. Organization of this paper
  5. Notation and conventions
  6. Warming up
  7. Preliminaries
  8. A tour of club-guessing
  9. Coherent sequences
  10. When coherence is not available
  11. Partitioned the club-guessing
  12. Using colourings
  13. Maintaining coherence
  14. Applications
  15. Increasing $\sigma$
  16. ...and 1 more sections

Key Result

Theorem A

For every successor cardinal $\lambda$, there exists a $\lambda$-bounded $C$-sequence $\vec{C}=\langle C_\delta\mathrel{|} \delta\in E^{\lambda^+}_\lambda\rangle$ satisfying the following. For every club $D \subseteq \lambda^+$, there is a $\delta \in E^{\lambda^+}_\lambda$ such that the following s

Theorems & Definitions (140)

  • Definition 1.8
  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem D
  • Theorem E
  • Definition 2.1: paper22
  • Definition 2.2: Main definition
  • Proposition 2.4
  • Definition 2.5
  • ...and 130 more