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Fragments of Martin's axiom

Yinhe Peng

TL;DR

This work analyzes the landscape of fragments of Martin's Axiom around Knaster-type properties. It proves the striking result that MA for ω_1-dense sets is equivalent to the fragment asserting K_3 for every ccc poset, and it shows that the dimension 3 in K_n is in a precise sense minimal. Through colorings, poset constructions from towers, and a general damage-control forcing framework, the authors relate Ramsey-type fragments, precaliber notions, and σ-linked properties to MA_ω1, establishing both positive implications (e.g., 𝒦_3⇒MA_ω1) and separations between fragments (e.g., P_{ω1}(K_n→σ-n-linked) vs MA_ω1(K_n)). The methodology includes coding finitary colorings by higher-arity colorings, forcing with colorings to control centered subsets, and iterative constructions that preserve ω_1 while separating fragments. These results clarify the strength and hierarchy of forcing axioms near MA_ω1 and provide toolkits (coding, damage control) potentially applicable to broader cardinal settings.

Abstract

We show that Martin's axiom for $ω_1$ dense sets is equivalent to its fragment asserting that every ccc poset has the Knaster property K$_3$. On the other hand, we show that the dimension 3 in K$_3$ is in some sense minimal.

Fragments of Martin's axiom

TL;DR

This work analyzes the landscape of fragments of Martin's Axiom around Knaster-type properties. It proves the striking result that MA for ω_1-dense sets is equivalent to the fragment asserting K_3 for every ccc poset, and it shows that the dimension 3 in K_n is in a precise sense minimal. Through colorings, poset constructions from towers, and a general damage-control forcing framework, the authors relate Ramsey-type fragments, precaliber notions, and σ-linked properties to MA_ω1, establishing both positive implications (e.g., 𝒦_3⇒MA_ω1) and separations between fragments (e.g., P_{ω1}(K_n→σ-n-linked) vs MA_ω1(K_n)). The methodology includes coding finitary colorings by higher-arity colorings, forcing with colorings to control centered subsets, and iterative constructions that preserve ω_1 while separating fragments. These results clarify the strength and hierarchy of forcing axioms near MA_ω1 and provide toolkits (coding, damage control) potentially applicable to broader cardinal settings.

Abstract

We show that Martin's axiom for dense sets is equivalent to its fragment asserting that every ccc poset has the Knaster property K. On the other hand, we show that the dimension 3 in K is in some sense minimal.
Paper Structure (23 sections, 77 theorems, 191 equations, 3 figures)

This paper contains 23 sections, 77 theorems, 191 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Strengthening ccc to property K$_n$
  3. Fragments above property K
  4. Ramsey-type fragments
  5. Contents and organization
  6. Preliminary
  7. Constructing colorings and posets from a tower
  8. $\mathscr{K}_3$ implies $\mathfrak{t}>\omega_1$
  9. P($\sigma$-$n$-linked$\rightarrow$K$_{n+1}$) implies $\mathfrak{t}>\omega_1$
  10. P$_{\omega_1}$(precaliber $\omega_1\rightarrow \sigma$-linked) implies $\mathfrak{b}>\omega_1$
  11. P$_{\omega_1}$(precaliber $\omega_1\rightarrow \sigma$-3-linked) implies $\mathfrak{t}>\omega_1$
  12. Coding finatary coloring by $(n+1)$-ary coloring
  13. Coding a poset by an uncountable homogeneous set
  14. P($\sigma$-$n$-linked$\rightarrow$precaliber $\omega_1$) implies P$_{\omega_1}$($\sigma$-$n$-linked$\rightarrow \sigma$-centered)
  15. P(K$_n\rightarrow$K$_{n+1}$) implies MA$_{\omega_1}$(K$_n$)
  16. ...and 8 more sections

Key Result

Theorem 1

$\mathscr{K}_3$ implies MA$_{\omega_1}$.

Figures (3)

  • Figure 1: Forcing properties
  • Figure 2: Forcing properties' strength
  • Figure 3: Weak forcing properties

Theorems & Definitions (170)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Definition 1
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Definition 2
  • Definition 3
  • Definition 4
  • ...and 160 more