  We show how to adapt the Hardy--Littlewood circle method to count
monochromatic solutions to diagonal Diophantine equations. This delivers a
lower bound which is optimal up to absolute constants. The method is
illustrated on equations obtained by setting a diagonal quadratic form equal to
a linear form. As a consequence, we determine an algebraic criterion for when
such equations are partition regular. Our methods involve discrete harmonic
analysis and require a number of `mixed' restriction estimates, which may be of
independent interest.
