Rado's theorem (Ramsey theory)

Rado's theorem is a theorem from the branch of mathematics known as Ramsey theory. It is named for the German mathematician Richard Rado. It was proved in his thesis, Studien zur Kombinatorik.

Let ${\displaystyle A\mathbf {x} =\mathbf {0} }$ be a system of linear equations, where ${\displaystyle A}$ is a matrix with integer entries. This system is said to be ${\displaystyle r}$-regular if, for every ${\displaystyle r}$-coloring of the natural numbers 1, 2, 3, ..., the system has a monochromatic solution. A system is regular if it is r-regular for all r ≥ 1.

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