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Strong partition relations below the power set: consistency, was Sierpinski right, II?

Saharon Shelah

Abstract

We continue here [She88] but we do not rely on it. The motivation was a conjecture of Galvin stating that 2^{omega} >= omega_2 + omega_2-> [omega_1]^{n}_{h(n)} is consistent for a suitable h: omega-> omega. In section 5 we disprove this and give similar negative results. In section 3 we prove the consistency of the conjecture replacing omega_2 by 2^omega, which is quite large, starting with an Erdős cardinal. In section 1 we present iteration lemmas which are needed when we replace omega by a larger lambda and in section 4 we generalize a theorem of Halpern and Lauchli replacing omega by a larger lambda .

Strong partition relations below the power set: consistency, was Sierpinski right, II?

Abstract

We continue here [She88] but we do not rely on it. The motivation was a conjecture of Galvin stating that 2^{omega} >= omega_2 + omega_2-> [omega_1]^{n}_{h(n)} is consistent for a suitable h: omega-> omega. In section 5 we disprove this and give similar negative results. In section 3 we prove the consistency of the conjecture replacing omega_2 by 2^omega, which is quite large, starting with an Erdős cardinal. In section 1 we present iteration lemmas which are needed when we replace omega by a larger lambda and in section 4 we generalize a theorem of Halpern and Lauchli replacing omega by a larger lambda .

Paper Structure

This paper contains 6 sections, 10 theorems, 48 equations.

Table of Contents

  1. Preliminaries
  2. Introduction
  3. Introducing the partition on trees
  4. Consistency of a strong partition below the continuum
  5. Partition theorem for trees on large cardinals
  6. Somewhat complementary negative partition relation in ZFC

Key Result

Lemma 1.3

If $\mathbf{q} = \langle \mathbb{P}_i,\,\mathbb{Q}_j : i \le i(*),\ j < i(*)\rangle$ is a $(<\mu)$-support iteration and $(a)$ or $(b)$ or $(c)$ below hold, then it is a $*_\mu^\varepsilon$-iteration.

Theorems & Definitions (65)

  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 1.1
  • Definition 1.2
  • Lemma 1.3
  • proof : Proof.
  • Theorem 1.4
  • Remark 1.6
  • Definition 1.7
  • ...and 55 more