  We obtain a double exponential bound in Brauer's generalisation of van der
Waerden's theorem, which concerns progressions with the same colour as their
common difference. Such a result has been obtained independently and in much
greater generality by Sanders. Using Gowers' local inverse theorem, our bound
is quintuple exponential in the length of the progression. We refine this bound
in the colour aspect for three-term progressions, and combine our arguments
with an insight of Lefmann to obtain analogous bounds for the Ramsey numbers of
certain nonlinear quadratic equations.
