TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-11

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Schur's theorem: every finite colouring of a sufficiently long positive initial interval $\{1,\ldots,N\}$ 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 $\{1,\ldots,N\}$ 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 $\mathbb{N}$ (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 $N\to(r)^k_c$ .

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 $\{0,\ldots,M-1\}$ , 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 · next 3 levels

Direct dependencies and their dependencies through the next three levels: 37 results over 20 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click 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,\ldots,N\}{1,…,N} has positive monochromatic x,y,zx,y,zx,y,z with x+y=zx+y=zx+y=z