Geodesic
Minimal definable graphs of definable chromatic number at least three
Forum of Mathematics, Sigma, Tome 9 (2021)

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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. 
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     title = {Minimal definable graphs of definable chromatic number at least three},
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Raphaël Carroy; Benjamin D. Miller; David Schrittesser; Zoltán Vidnyánszky. Minimal definable graphs of definable chromatic number at least three. Forum of Mathematics, Sigma, Tome 9 (2021). doi: 10.1017/fms.2020.58

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