## Schur's Lemma

For each there exists a largest Integer (known as the Schur Number) such that no matter how the set of Integers less than (where is the Floor Function) is partitioned into classes, one class must contain Integers , , such that , where and are not necessarily distinct. The upper bound has since been slightly improved to .

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.