Schur's theorem

In discrete mathematics, Schur's theorem is any of several theorems of the mathematician Issai Schur. In differential geometry, Schur's theorem is a theorem of Axel Schur. In functional analysis, Schur's theorem is often called Schur's property, also due to Issai Schur. In Ramsey theory, Schur's theorem states that for every positive integer c, there exists a positive integer S, such that for every partition of the integers { 1 , … , S } {\displaystyle \{1,\ldots ,S\}} into c parts, one of the parts contains integers x, y and z with x + y = z {\displaystyle x+y=z} . For every positive integer c, S(c) denotes the smallest number S such that for every partition of the integers { 1 , … , S } {\displaystyle \{1,\ldots ,S\}} into c parts, one of the parts contains integers x, y, and z with x + y = z {\displaystyle x+y=z} . Schur's theorem ensures that S(c) is well-defined for every positive integer c. The numbers of the form S(c) are called Schur's numbers. Folkman's theorem generalizes Schur's theorem by stating that there exist arbitrarily large sets of integers, all of whose nonempty sums belong to the same part. Using this definition, the only known Schur numbers are S(n) = 2, 5, 14, 45, and 161 (sequence A030126 in the OEIS). The proof that S(5) = 161 was announced in 2017 and required 2 petabytes of space. In combinatorics, Schur's theorem tells the number of ways for expressing a given number as a (non-negative, integer) linear combination of a fixed set of relatively prime numbers.