Abstract :
A u×v matrix A is image partition regular provided that, whenever N is finitely colored, there is some x→∈Nv with all entries of Ax→ monochrome. Image partition regular matrices are a natural way of representing some of the classic theorems of Ramsey Theory, including theorems of Hilbert, Schur, and van der Waerden. We present here some new characterizations and consequences of image partition regularity and investigate some issues raised by these. One of our characterizations is that the image partition regular matrices are precisely those that preserve a certain notion of largeness (“central sets”)—we examine what happens for other well known notions of largeness. Another property of image partition regular matrices is that (except in trivial cases) the entries of Ax→ may be chosen to be distinct—we investigate when we may choose the entries to be “close together” or “far apart” in various senses.
