A+ CATEGORY SCIENTIFIC UNIT

Coloring ordinals by reals

Volume 196 / 2007

Jörg Brendle, Sakaé Fuchino Fundamenta Mathematicae 196 (2007), 151-195 MSC: 03E05, 03E17, 03E35, 03E65. DOI: 10.4064/fm196-2-5

Abstract

We study combinatorial principles we call the Homogeneity Principle ${\rm HP}(\kappa)$ and the Injectivity Principle ${\rm IP}(\kappa,\lambda)$ for regular $\kappa>\aleph_1$ and $\lambda\leq\kappa$ which are formulated in terms of coloring the ordinals $<\kappa$ by reals.

These principles are strengthenings of ${\rm C}^{\rm s}(\kappa)$ and ${\rm F}^{\rm s}(\kappa)$ of I. Juhász, L. Soukup and Z. Szentmiklóssy. Generalizing their results, we show e.g. that ${\rm IP}(\aleph_2,\aleph_1)$ (hence also ${\rm IP}(\aleph_2,\aleph_2)$ as well as ${\rm HP}(\aleph_2)$) holds in a generic extension of a model of CH by Cohen forcing, and ${\rm IP}(\aleph_2,\aleph_2)$ (hence also ${\rm HP}(\aleph_2)$) holds in a generic extension by countable support side-by-side product of Sacks or Prikry–Silver forcing (Corollary 4.8). We also show that the latter result is optimal (Theorem 5.2).

Relations between these principles and their influence on the values of the variations $\mathfrak b^\uparrow$, $\mathfrak b^h$, $\mathfrak b^*$, $\mathfrak {do}$ of the bounding number $\mathfrak b$ are studied.

One of the consequences of ${\rm HP}(\kappa)$ besides ${\rm C^s}(\kappa)$ is that there is no projective well-ordering of length $\kappa$ on any subset of ${}^{\omega}\omega$. We construct a model in which there is no projective well-ordering of length $\omega_2$ on any subset of ${}^{\omega}\omega$ ($\mathfrak{do}=\aleph_1$ in our terminology) while $\mathfrak b^*=\aleph_2$ (Theorem 6.4).

Authors

  • Jörg BrendleDepartment of Computer Science
    and Systems Engineering
    Graduate School of Engineering
    Kobe University
    Rokko-dai 1-1
    Nada, Kobe 657-8501, Japan
    e-mail
  • Sakaé FuchinoDepartment of Natural Science
    and Mathematics
    College of Engineering, Chubu University
    Kasugai, Aichi 487-8501, Japa
    e-mail

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