Lesson 5

We present (without proof) a similar result.

Theorem (Schur, 1917). For any \(k \in \mathbb{Z}_{\ge 2}\), there exists a number \(n \in \mathbb{Z}\) such that in any coloring of the elements of the set \([n]\) into \(k\) colors, there exist three elements \(i,j,k\) of the same color for which \(i+j=k\).