Schur's theorem

Remarks

Let $N + 1$ be at least the Ramsey number for $r$ colours and triangles, given by Ramsey's theorem. Colour the edge $\set{i, j}$ of the complete graph on $\set{1, \ldots, N + 1}$ with the colour of $\abs{i - j}$. A monochromatic triangle $i < j < k$ gives $x = j - i$, $y = k - j$ and $z = k - i$ of the same colour with $x + y = z$.

The least such $N$ is $S(r) + 1$, where the Schur number $S(r)$ is the largest $n$ for which $\set{1, \ldots, n}$ can be split into $r$ sum-free sets. For example $S(2) = 4$: the colouring $\set{1, 4}$, $\set{2, 3}$ has no monochromatic solution, while every $2$-colouring of $\set{1, \ldots, 5}$ has one. Known values are $S(1) = 1$, $S(2) = 4$, $S(3) = 13$, $S(4) = 44$ and $S(5) = 160$.

Going deeper

For a student

Schur proved the theorem in 1916 while studying Fermat's equation modulo primes. It implies that for each $n$ and all large enough primes $p$, the congruence $x^n + y^n \equiv z^n \pmod p$ has a solution with $xyz \not\equiv 0$. The proof is a direct reduction to Ramsey's theorem for triangles. Rado's theorem extends it to all linear equations whose monochromatic solutions are forced in this way.