Image partition regularity of affine transformations

If u , v ∈ N , A is a u × v matrix with entries from Q , and b → ∈ Q u , then ( A , b → ) determines an affine transformation from Q v to Q u by x → ↦ A x → + b → . In 1933 and 1943 Richard Rado determined precisely when such transformations are kernel partition regular over N , Z , or Q , meaning that whenever the nonzero elements of the relevant set are partitioned into finitely many cells, there is some element of the kernel of the transformation with all of its entries in the same cell. In 1993 the first author and Imre Leader determined when such transformations with b → = 0 ¯ are image partition regular over N , meaning that whenever N is partitioned into finitely many cells, there is some element of the image of the transformation with all of its entries in the same cell. In this paper we characterize the image partition regularity of such transformations over N , Z , or Q for arbitrary b → .