# Measurable chromatic numbers

@article{Miller2008MeasurableCN, title={Measurable chromatic numbers}, author={Benjamin D. Miller}, journal={Journal of Symbolic Logic}, year={2008}, volume={73}, pages={1139 - 1157} }

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]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…

## 9 Citations

Borel chromatic numbers of graphs of commuting functions

- MathematicsFundamenta Mathematicae
- 2021

Let D = (X,D) be a Borel directed graph on a standard Borel space X and let χB(D) be its Borel chromatic number. If F0, . . . , Fn−1 : X → X are Borel functions, let DF0,...,Fn−1 be the directed…

Definable Combinatorics of Graphs and Equivalence Relations

- Mathematics
- 2018

Let D = (X, D) be a Borel directed graph on a standard Borel space X and let χB(D) be its Borel chromatic number. If F0, …, Fn-1: X → X are Borel functions, let DF0, …, Fn-1 be the directed graph…

An antibasis result for graphs of infinite Borel chromatic number

- Mathematics
- 2014

A graph on a set X is a symmetric, irreflexive subset of X ×X. For a graph G on X, we let degG(x) = |{y ∈ X : (x, y) ∈ G}|. If degG(x) is countable for all x ∈ X we say that G is locally countable.…

Borel Colouring Bad Rays

- MathematicsOrder
- 2022

Every better quasi-order codifies a Borel graph that does not contain a copy of the shift graph. It is known that there is a better quasi-order that codes a Borel graph with infinite Borel chromatic…

Measurable chromatic and independence numbers for ergodic graphs and group actions

- Mathematics
- 2013

We study in this paper combinatorial problems concerning graphs generated by
measure preserving actions of countable groups on standard measure spaces. In particular
we study chromatic and…

A Cantor--Bendixson dichotomy of domatic partitions

- Mathematics
- 2022

Let Γ = ∏ i ∈ N Γ i be an inﬁnite product of nontrivial ﬁnite groups, or let Γ = ( R / Z ) n be a ﬁnite-dimensional torus. For a countably inﬁnite set S ⊆ Γ , an ℵ 0 -domatic partition is a partial ℵ…

Weak Borel chromatic numbers

- MathematicsMath. Log. Q.
- 2011

It is consistent with an arbitrarily large size of the continuum that every closed graph on a Polish space either has a perfect clique or has a weak Borel chromatic number of at most ℵ1.

MEASURABLE 3-COLORINGS OF ACYCLIC GRAPHS

- Mathematics
- 2010

This is the first of two lectures on measurable chromatic numbers given in June 2010 at the University of Barcelona. Our main result here is that acyclic locally finite analytic graphs on Polish…

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