Schur's Lemma

For each $k\in\Bbb{N}$ there exists a largest Integer $s(k)$ (known as the Schur Number) such that no matter how the set of Integers less than $\left\lfloor{n!e}\right\rfloor$ (where $\left\lfloor{x}\right\rfloor$ is the Floor Function) is partitioned into $k$ classes, one class must contain Integers $x$, $y$, $z$ such that $x+y=z$, where $x$ and $y$ are not necessarily distinct. The upper bound has since been slightly improved to $\left\lfloor{n!(e-1/24)}\right\rfloor$.

See also Combinatorics, Schur Number, Schur's Theorem

References

Guy, R. K. ``Schur's Problem. Partitioning Integers into Sum-Free Classes'' and ``The Modular Version of Schur's Problem.'' §E11 and E12 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 209-212, 1994.