The Schur number S(k) is the largest integer n for which the interval [1, n] can be partitioned into k sum-free sets. S(k) is guaranteed to exist for each k by Schur's problem. Note the definition of the Schur number as the smallest number S'(k) = S(k) + 1 for which such a partition does not exist is also prevalent in the literature (OEIS A030126). Schur gave the lower bound S(k)>=1/2(3^n - 1) which is sharp for n = 1, 2, and 3.

Ramsey number | Ramsey's theorem | Schur's problem | Schur's Ramsey theorem