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.
Monochromatic products and sums in $2$-colorings of $\mathbb{N}$
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abstract
We show that any $2$-coloring of $\mathbb{N}$ contains infinitely many monochromatic sets of the form $\{x,y,xy,x+y\},$ and more generally monochromatic sets of the form $\{x_i,\prod x_i,\sum x_i: i\leq k\}$ for any $k\in\mathbb{N}.$ Along the way we prove a monochromatic products of sums theorem that extends Hindman's theorem and a colorful variant of this result that holds in any 'balanced' coloring.
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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.