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

Minimal definable graphs of definable chromatic number at least three