van der Waerden's Theorem

For any given Positive Integers $k$ and $r$, there exists a threshold number $n(k,r)$ (known as a van der Waerden Number) such that no matter how the numbers 1, 2, ..., $n$ are partitioned into $k$ classes, at least one of the classes contains an Arithmetic Progression of length at least $r$. However, no Formula for $n(k,r)$ is known.

See also Arithmetic Progression

References

Honsberger, R. More Mathematical Morsels. Washington, DC: Math. Assoc. Amer., p. 29, 1991.

Khinchin, A. Y. ``Van der Waerden's Theorem on Arithmetic Progressions.'' Ch. 1 in Three Pearls of Number Theory. New York: Dover, pp. 11-17, 1998.

van der Waerden, B. L. ``Beweis einer Baudetschen Vermutung.'' Nieuw Arch. Wiskunde 15, 212-216, 1927.