Coloring ordinals by reals

Jörg Brendle¹ ; Sakaé Fuchino²

¹ Department of Computer Science and Systems Engineering Graduate School of Engineering Kobe University Rokko-dai 1-1 Nada, Kobe 657-8501, Japan
² Department of Natural Science and Mathematics College of Engineering, Chubu University Kasugai, Aichi 487-8501, Japa

Fundamenta Mathematicae, Tome 196 (2007) no. 2, pp. 151-195

Résumé

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).

Détail

DOI : 10.4064/fm196-2-5

Keywords: study combinatorial principles call homogeneity principle kappa injectivity principle kappa lambda regular kappa aleph lambda leq kappa which formulated terms coloring ordinals kappa reals these principles strengthenings kappa kappa nbsp juh nbsp soukup nbsp szentmikl ssy generalizing their results aleph aleph hence aleph aleph aleph holds generic extension model cohen forcing aleph aleph hence aleph holds generic extension countable support side by side product sacks prikry silver forcing corollary nbsp latter result optimal theorem nbsp relations between these principles their influence values variations mathfrak uparrow mathfrak mathfrak * mathfrak bounding number mathfrak studied consequences kappa besides kappa there projective well ordering length kappa subset omega omega construct model which there projective well ordering length omega subset omega omega mathfrak aleph terminology while mathfrak * aleph theorem nbsp