Density methods for partition regularity

A system of linear equations with integer coefficients is partition regular if, whenever the natural numbers are finitely coloured, there is a monochromatic solution. The finite partition regular systems were completely characterised by Rado in terms of a simple property of their matrix of coefficients. As a result, finite partition regular systems are very well understood.

Much less is known about infinite systems. In fact, only a very few families of infinite partition regular systems are known. I'll explain a relatively new method of constructing infinite partition regular systems, and describe how it has been applied to settle some basic questions in the area.