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 . 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 any partition of the positive integers into a finite number of parts, one of the parts contains three integers x, y, z with

Moreover, for every positive integer c, there exists a number S(c), called Schur's number, such that for every partition of the integers

into c parts, one of the parts contains integers x, y, and z with

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.

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. In particular, if ${\displaystyle \{a_{1},\ldots ,a_{n}\}}$ is a set of integers such that ${\displaystyle \gcd(a_{1},\ldots ,a_{n})=1}$, the number of different tuples of non-negative integer numbers ${\displaystyle (c_{1},\ldots ,c_{n})}$ such that ${\displaystyle x=c_{1}a_{1}+\cdots +c_{n}a_{n}}$ when ${\displaystyle x}$ goes to infinity is:

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