Journal of Symbolic Logic

Measurable chromatic numbers

Benjamin D. Miller

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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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J. Symbolic Logic, Volume 73, Issue 4 (2008), 1139-1157.

First available in Project Euclid: 27 December 2008

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Miller, Benjamin D. Measurable chromatic numbers. J. Symbolic Logic 73 (2008), no. 4, 1139--1157. doi:10.2178/jsl/1230396910.

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