Bootstrapping partition regularity of linear systems

Suppose that A is a k x d matrix of integers and write R-A : N -> N boolean OR {infinity} for the function taking r to the largest N such that there is an r-colouring C of [N] with boolean OR(C subset of C) C-d boolean AND ker A = empty set. We show that if R-A(r) < infinity for all r is an element of N then R-A(r) <= exp(exp(r(O)A((1)))) for all r >= 2. When the kernel of A consists only of Brauer configurations - that is, vectors of the form (y, x, x + y, ..., x + (d - 2)y) - the above statement has been proved by Chapman and Prendiville with good bounds on the O-A(1) term.