Coloring ordinals by reals

Abstract

We study combinatorial principles we call the Homogeneity Principle HP(κ) and the Injectivity Principle IP(κ, λ) for regular κ > N 1 and λ ≤ κ which are formulated in terms of coloring the ordinals < κ by reals. These principles are strengthenings of C s(κ) and F s(κ) of I. Juhász, L. Soukup and Z. Szentmiklóssy. Generalizing their results, we show e.g. that IP(N 2, N 1) (hence also IP(N 2, N 2) as well as HP(N 2)) holds in a generic extension of a model of CH by Cohen forcing, and IP(N 2, N 2) (hence also HP(N 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 b†, b h, b*, o of the bounding number b are studied. One of the consequences of HP(κ) besides C s(κ) is that there is no projective well-ordering of length κ on any subset of ωω. We construct a model in which there is no projective well-ordering of length ω 2 on any subset of ωω (o = N 1 in our terminology) while b* = N 2 (Theorem 6.4). © Instytut Matematyczny PAN, 2007.