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