On Wikipedia page on Rado's theorem, it says that the system $Ax=0$ is said to be $r-regular$ if for every $r$ coloring of the natural numbers, the system has a monochromatic solution. I don't understand what they mean by solution. Here $x$ is a vector in $Ax=0$, so the solution is not an integer.
Context: Rado's theorem can be used to prove Van der Warden's theorem. And the van der Waerden theorem and Szemeredi's regularity lemma can be used to prove szemeredi's theorem which can be used to prove relative Szemeredi's theorem and can then be used to proof Green-Tao's theorem.