Coloring ordinals by reals
Volume 196 / 2007
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).
