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.
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Ramsey theory
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.
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
As a result, for every set of relatively prime numbers
Differential geometry
In differential geometry, Schur's theorem compares the distance between the endpoints of a space curve
Suppose
Schur's theorem is usually stated for
Linear algebra
In linear algebra Schur’s theorem is referred to as either the triangularization of a square matrix with complex entries, or of a square matrix with real entries and real eigenvalues.
Functional analysis
In functional analysis and the study of Banach spaces, Schur's theorem, due to J. Schur, often refers to Schur's property, that for certain spaces, 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 coefficients, if S is the set of all nonzero values