TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-11

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Schur's theorem: every finite colouring of a sufficiently long positive initial interval ${1, \dots, N}$ has positive monochromatic $x, y, z$ with $x + y = z$

Statement

For every positive number $c$ of colours there is a positive natural $N$ such that every $c$ -colouring of ${1, \dots, N}$ has positive $x, y, z$ of one colour satisfying $x + y = z$ . The variables need not be distinct. Natural order is that of The natural numbers $N$ (von Neumann) and Order on the natural numbers, and the proof uses the pair-colouring convention of Finite colourings of $k$ -element subsets, monochromatic sets, and the arrow notations $N \to (s, t)^{2}$ and $N \to (r)_{c}^{k}$ .

Facts & Assumptions

Given: A positive number $c$ of colours and a colouring of a sufficiently long positive initial interval.

[L1]

For all positive $s, t$ , $(\binom{s + t - 2}{s - 1}) \to (s, t)^{2}$ (Finite graph Ramsey theorem: $(\binom{s + t - 2}{s - 1}) \to (s, t)^{2}$ for all positive $s, t$ ).

Proof

technique · direct

1.1

Iterating [L1] gives a finite $M$ such that every $c$ -colouring of the pairs of an $M$ -element set has a monochromatic triangle: separate one colour from the remaining colours, use [L1] with target $3$ for the first colour and with a recursively chosen target for the others, and continue through the finite colour list.

2.1

Colour the edge ${i, j}$ of the ordered vertex set ${0, \dots, M - 1}$ , with $i < j$ , by the given colour of the positive difference $j - i$ . Step 1.1 gives a monochromatic triangle $i < j < k$ .

step 1.1

3.1

Put $x = j - i$ , $y = k - j$ and $z = k - i$ . These are positive, the edge colouring says they have one original colour, and arithmetic gives $x + y = z$ . Taking $N = M - 1$ contains all three differences.

step 2.1algebra∎

Depends on

Used by

The Schur number S(c) as the largest N for which {1,…,N} has a c-colouring with no positive monochromatic solution of x+y=z Definition

Dependency tree · two levels

16 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

I. B. Leader, Ramsey Theory, remark after Theorem 8 (standard reference, not scraped)

Schur's theorem: every finite colouring of a sufficiently long positive initial interval {1,…,N}{1,…,N} has positive monochromatic x,y,zx,y,z with x+y=zx+y=z