Partition regularity of polynomial equations over Z is undecidable if Hilbert's tenth problem over Q is undecidable, and over function fields it is unconditionally Pi_2^0-complete.
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Undecidability in the Ramsey theory of polynomial equations and Hilbert's tenth problem
Partition regularity of polynomial equations over Z is undecidable if Hilbert's tenth problem over Q is undecidable, and over function fields it is unconditionally Pi_2^0-complete.