We study combinatorial principles we call the Homogeneity Principle HP(κ) and the Injectivity Principle IP(κ,λ) for regular κ > ℵ₁ and λ ≤ κ which are formulated in terms of coloring the ordinals < κ by reals. These principles are strengthenings of and of I. Juhász, L. Soukup and Z. Szentmiklóssy. Generalizing their results, we show e.g. that IP(ℵ₂,ℵ₁) (hence also IP(ℵ₂,ℵ₂) as well as HP(ℵ₂)) holds in a generic extension of a model of CH by Cohen forcing, and IP(ℵ₂,ℵ₂) (hence also HP(ℵ₂)) 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 , , *, of the bounding number are studied. One of the consequences of HP(κ) besides is that there is no projective well-ordering of length κ on any subset of . We construct a model in which there is no projective well-ordering of length ω₂ on any subset of ( = ℵ₁ in our terminology) while * = ℵ₂ (Theorem 6.4).
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm196-2-5,
author = {J\"org Brendle and Saka\'e Fuchino},
title = {Coloring ordinals by reals},
journal = {Fundamenta Mathematicae},
volume = {193},
year = {2007},
pages = {151-195},
zbl = {1143.03023},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm196-2-5}
}
Jörg Brendle; Sakaé Fuchino. Coloring ordinals by reals. Fundamenta Mathematicae, Tome 193 (2007) pp. 151-195. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm196-2-5/