The Borel directed graphs with dichromatic number at least 3 form a Π¹₂-complete class, so no countable basis exists under Borel homomorphisms.
An algebraic approach to Borel CSPs
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No Countable Basis for Borel Directed Graphs of Dichromatic Number at Least Three
The Borel directed graphs with dichromatic number at least 3 form a Π¹₂-complete class, so no countable basis exists under Borel homomorphisms.