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Coloring ordinals by reals

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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 $C^{s}(κ)$ and $F^{s}(κ)$ 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 $𝔟^{↑}$, $𝔟^{h}$, 𝔟*, 𝔡𝔬 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 $^{ω}ω$. We construct a model in which there is no projective well-ordering of length ω₂ on any subset of $^{ω}ω$ (𝔡𝔬 = ℵ₁ in our terminology) while 𝔟* = ℵ₂ (Theorem 6.4).
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  • Department of Computer Science and Systems Engineering, Graduate School of Engineering, Kobe University, Rokko-dai 1-1, Nada, Kobe 657-8501, Japan
  • Department of Natural Science and Mathematics, College of Engineering, Chubu University, Kasugai, Aichi 487-8501, Japa
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bwmeta1.element.bwnjournal-article-doi-10_4064-fm196-2-5
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