Rado's theorem (Ramsey theory)

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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 Ax = 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 Ax=0 is regular if and only if the matrix A satisfies the columns condition. Let ci denote the i-th column of A. The matrix A satisfies the columns condition provided that there exists a partition C1, C2, ..., Cn of the column indices such that if s_i = \Sigma_{j \in C_i}c_j, then

  1. s1 = 0
  2. for all i ≥ 2, si can be written as a rational[1] linear combination of the cj's in the Ck 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

x_T = \sum_{i\in T}x_{\{i\}},

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


  1. ^ Modern graph theory by Béla Bollobás. 1st ed. 1998. ISBN 978-0-387-98488-9. Page 204
  2. ^ Graham, Ronald L.; Rothschild, Bruce L.; Spencer, Joel H. (1980), "3.4 Finite Sums and Finite Unions (Folkman's Theorem)", Ramsey Theory, Wiley-Interscience, pp. 65–69 .