  A finite or infinite matrix $A$ with rational entries (and only finitely many
non-zero entries in each row) is called image partition regular if, whenever
the natural numbers are finitely coloured, there is a vector $x$, with entries
in the natural numbers, such that $Ax$ is monochromatic. Many of the classical
results of Ramsey theory are naturally stated in terms of image partition
regularity.
  Our aim in this paper is to investigate maximality questions for image
partition regular matrices. When is it possible to add rows on to $A$ and
remain image partition regular? When can one add rows but `nothing new is
produced'? What about adding rows and also new variables? We prove some results
about extensions of the most interesting infinite systems, and make several
conjectures.
  Perhaps our most surprising positive result is a compatibility result for
Milliken-Taylor systems, stating that (in many cases) one may adjoin one
Milliken-Taylor system to a translate of another and remain image partition
regular. This is in contrast to earlier results, which had suggested a strong
inconsistency between different Milliken-Taylor systems. Our main tools for
this are some algebraic properties of the $\beta N$, the Stone-Cech
compactification of the natural numbers.
