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We show how multiplicatively syndetic sets can be used in the study ofpartition regularity of dilation invariant systems of polynomial equations. Inparticular, we prove that a dilation invariant system of polynomial equationsis partition regular if and only if it has a solution inside everymultiplicatively syndetic set. We also adapt the methods of Green-Tao andChow-Lindqvist-Prendiville to develop a syndetic version of Roth's densityincrement strategy. This argument is then used to obtain bounds on the Radonumbers of configurations of the form $\{x, d, x + d, x + 2d\}$.

Partition regularity and multiplicatively syndetic sets