Editing Schur's theorem

{{short description|One of several theorems in different areas of mathematics}}
In [[discrete mathematics]], '''Schur's theorem''' is any of several [[theorem]]s of the [[mathematician]] [[Issai Schur]]. In [[differential geometry]], '''Schur's theorem''' is a theorem of [[:de:Axel Schur|Axel Schur]]. In [[functional analysis]], '''Schur's theorem''' is often called [[Schur's property]], also due to Issai Schur.

== Ramsey theory ==
{{wikibooks|Combinatorics|Schur's Theorem|Proof of Schur's theorem}}
In [[Ramsey theory]], '''Schur's theorem''' states that for any [[Partition of a set|partition]] of the [[positive integer]]s into a finite number of parts, one of the parts contains three integers ''x'', ''y'', ''z'' with 

:<math>x + y = z.</math>

For every positive integer ''c'', ''S''(''c'') denotes the smallest number ''S'' such that for every partition of the integers <math>\{1,\ldots, S \}</math> into ''c'' parts, one of the parts contains integers ''x'', ''y'', and ''z'' with <math>x + y = z</math>. 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 number'''s.

[[Folkman's theorem]] generalizes Schur's theorem by stating that there exist arbitrarily large sets of integers, all of whose [[empty sum|nonempty]] sums belong to the same part.

Using this definition, the only known Schur numbers are ''S''(n) {{=}} 2, 5, 14, 45, and 161 ({{oeis|A030126}}) The [[mathematical proof|proof]] that {{nobr|''S''(5) {{=}} 161}} was announced in 2017 and required 2 [[petabyte]]s of space.<ref>{{cite arXiv |last=Heule |first=Marijn J. H. |author-link=Marijn Heule|title=Schur Number Five |eprint=1711.08076 |date=2017 |class=cs.LO }}</ref><ref>{{Cite web|title=Schur Number Five|url=https://www.cs.utexas.edu/~marijn/Schur/|access-date=2021-10-06|website=www.cs.utexas.edu}}</ref>

== Combinatorics ==
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 <math>\{a_1,\ldots,a_n\}</math> is a set of integers such that <math>\gcd(a_1,\ldots,a_n)=1</math>, the number of different multiples of non-negative integer numbers <math>(c_1,\ldots,c_n)</math> such that <math>x=c_1a_1 + \cdots + c_na_n</math> when <math>x</math> goes to infinity is:

:<math>\frac{x^{n-1}}{(n-1)!a_1\cdots a_n}(1+o(1)).</math>

As a result, for every set of relatively prime numbers <math>\{a_1,\ldots,a_n\}</math> there exists a value of <math>x</math> such that every larger number is representable as a linear combination of <math>\{a_1,\ldots,a_n\}</math> in at least one way. This consequence of the theorem can be recast in a familiar context considering the problem of changing an amount using a set of coins. If the denominations of the coins are relatively prime numbers (such as 2 and 5) then any sufficiently large amount can be changed using only these coins. (See [[Coin problem]].)

== Differential geometry ==

{{main|Bow lemma}}

In [[differential geometry]], '''Schur's theorem''' compares the distance between the endpoints of a space curve <math>C^*</math> to the distance between the endpoints of a corresponding plane curve <math>C</math> of less curvature.

Suppose <math>C(s)</math> is a plane curve with curvature <math>\kappa(s)</math> which makes a convex curve when closed by the chord connecting its endpoints, and <math>C^*(s)</math> is a curve of the same length with curvature <math>\kappa^*(s)</math>. Let <math>d</math> denote the distance between the endpoints of <math>C</math> and <math>d^*</math> denote the distance between the endpoints of <math>C^*</math>. If <math>\kappa^*(s) \leq \kappa(s)</math> then <math>d^* \geq d</math>.

'''Schur's theorem''' is usually stated for <math>C^2</math> curves, but [[John M. Sullivan (mathematician)|John M. Sullivan]] has observed that Schur's theorem applies to curves of finite total curvature (the statement is slightly different).

==  Linear algebra ==
{{main|Schur decomposition}}
In [[linear algebra]], Schur’s theorem is referred to as either the [[Triangular_matrix#Triangularisability|triangularization]] of a [[square matrix]] with [[complex number|complex]] entries, or of a square matrix with [[real number|real]] entries and real [[eigenvalue]]s.

==Functional analysis==
In [[functional analysis]] and the study of [[Banach space]]s, Schur's theorem, due to [[I. Schur]], often refers to [[Schur's property]], that for certain spaces, [[weak topology|weak convergence]] implies convergence in the norm.

== Number theory ==
In [[number theory]], Issai Schur showed in 1912 that for every nonconstant [[polynomial]] ''p''(''x'') with integer [[coefficient]]s, if ''S'' is the set of all nonzero values <math>\begin{Bmatrix} p(n) \neq 0 : n \in \mathbb{N} \end{Bmatrix}</math>, then the set of [[prime number|primes]] that divide some member of ''S'' is infinite.

==See also==

*[[Schur's lemma (from Riemannian geometry)]]

==References==
{{reflist}}
* Herbert S. Wilf (1994). [http://www.cs.utsa.edu/~wagner/CS3343/resources/gfology.pdf generatingfunctionology]. Academic Press.
* [[Shiing-Shen Chern]] (1967). Curves and Surfaces in Euclidean Space. In ''Studies in Global Geometry and Analysis.'' Prentice-Hall.
* Issai Schur (1912). Über die Existenz unendlich vieler Primzahlen in einigen speziellen arithmetischen Progressionen, Sitzungsberichte der Berliner Math.

==Further reading==
* Dany Breslauer and Devdatt P. Dubhashi (1995). [http://www.brics.dk/LS/95/4/BRICS-LS-95-4/BRICS-LS-95-4.html Combinatorics for Computer Scientists]
* [[John M. Sullivan (mathematician)|John M. Sullivan]] (2006). [https://arxiv.org/abs/math.GT/0606007 Curves of Finite Total Curvature]. arXiv.

{{Functional analysis}}

[[Category:Theorems in discrete mathematics]]
[[Category:Ramsey theory]]
[[Category:Additive combinatorics]]
[[Category:Theorems in combinatorics]]
[[Category:Theorems in differential geometry]]
[[Category:Theorems in linear algebra]]
[[Category:Theorems in functional analysis]]
[[Category:Computer-assisted proofs]]