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December 2008 Measurable chromatic numbers
Benjamin D. Miller
J. Symbolic Logic 73(4): 1139-1157 (December 2008). DOI: 10.2178/jsl/1230396910

Abstract

We show that if add(null) = 𝔠, then the globally Baire and universally measurable chromatic numbers of the graph of any Borel function on a Polish space are equal and at most three. In particular, this holds for the graph of the unilateral shift on [ℕ], although its Borel chromatic number is ℵ₀. We also show that if add(null) = 𝔠, then the universally measurable chromatic number of every treeing of a measure amenable equivalence relation is at most three. In particular, this holds for “the” minimum analytic graph 𝔊₀ with uncountable Borel (and Baire measurable) chromatic number. In contrast, we show that for all κ ∈ { 2, 3, …, ℵ_0, 𝔠 }, there is a treeing of E₀ with Borel and Baire measurable chromatic number κ. Finally, we use a Glimm—Effros style dichotomy theorem to show that every basis for a non-empty initial segment of the class of graphs of Borel functions of Borel chromatic number at least three contains a copy of (ℝ<ℕ, ⊇).

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Benjamin D. Miller. "Measurable chromatic numbers." J. Symbolic Logic 73 (4) 1139 - 1157, December 2008. https://doi.org/10.2178/jsl/1230396910

Information

Published: December 2008
First available in Project Euclid: 27 December 2008

zbMATH: 1157.03025
MathSciNet: MR2467208
Digital Object Identifier: 10.2178/jsl/1230396910

Rights: Copyright © 2008 Association for Symbolic Logic

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Vol.73 • No. 4 • December 2008
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