For free Borel Z^d-grids, every grid has a Borel (2^d-1)-polychromatic coloring, while ergodic grids admit no Borel 2^d-polychromatic coloring, so the Borel threshold is 2^d-1.
Distributed algorithms, the lov´ asz local lemma, and descriptive combina- torics
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Borel Polychromatic Number of Grids
For free Borel Z^d-grids, every grid has a Borel (2^d-1)-polychromatic coloring, while ergodic grids admit no Borel 2^d-polychromatic coloring, so the Borel threshold is 2^d-1.