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 A x = 0 be a system of linear equations, where A is a matrix with integer entries. This system is said to be r -regular if, for every 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.

Rado's theorem states that a system A x = 0 is regular if and only if the matrix A satisfies the columns condition. Let c_i denote the i-th column of A. The matrix A satisfies the columns condition provided that there exists a partition C₁, C₂, ..., C_n of the column indices such that if s i = Σ j ∈ C i c j , then

1. s₁ = 0
2. for all i ≥ 2, s_i can be written as a rational linear combination of the c_j's in the C_k with k < i.

Folkman's theorem, the statement that there exist arbitrarily large sets of integers all of whose nonempty sums are monochromatic, may be seen as a special case of Rado's theorem concerning the regularity of the system of equations

where T ranges over each nonempty subset of the set {1, 2, ..., x}.

Other special cases of Rado's theorem are Schur's theorem and Van der Waerden's theorem.