# Minimal definable graphs of definable chromatic number at least three

@article{Carroy2019MinimalDG, title={Minimal definable graphs of definable chromatic number at least three}, author={Rapha{\"e}l Carroy and Benjamin D. Miller and David Schrittesser and Zolt{\'a}n Vidny{\'a}nszky}, journal={Forum of Mathematics, Sigma}, year={2019}, volume={9} }

Abstract We show that there is a Borel graph on a standard Borel space of Borel chromatic number three that admits a Borel homomorphism to every analytic graph on a standard Borel space of Borel chromatic number at least three. Moreover, we characterize the Borel graphs on standard Borel spaces of vertex-degree at most two with this property and show that the analogous result for digraphs fails.

## 4 Citations

### On Homomorphism Graphs

- MathematicsArXiv
- 2021

It is shown that for ∆ > 2 it is impossible to give a simple characterization of acyclic ∆-regular Borel graphs with Borel chromatic number at most ∆: such graphs form a Σ12-complete set, implying a strong failure of Brooks’-like theorems in the Borel context.

### MEASURABLE GRAPH COMBINATORICS

- Mathematics
- 2022

. We survey some recent results in the theory of measurable graph combinatorics. We also discuss applications to the study of hyperfiniteness and measurable equidecompositions.

### A complexity problem for Borel graphs

- MathematicsInventiones mathematicae
- 2021

We show that there is no simple (e.g. finite or countable) basis for Borel graphs with infinite Borel chromatic number. In fact, it is proved that the closed subgraphs of the shift graph on…

### $\Delta^1_1$ Effectivization in Borel Combinatorics

- Mathematics
- 2021

We develop a flexible method for showing that Borel witnesses to some combinatorial property of ∆11 objects yield ∆ 1 1 witnesses. We use a modification the Gandy–Harrington forcing method of proving…

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