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# van der Waerden's Theorem

van der Waerden's theorem is a theorem about the existence of arithmetic progressions in sets. The theorem can be stated in four equivalent forms.

1. If , then some contains arbitrarily long arithmetic progressions (Baudet's conjecture).

2. For all positive integers and , there exists a constant such that if and , then some set contains an arithmetic progression of length .

3. If is an infinite sequence of integers satisfying for some , then the sequence contains arbitrarily long arithmetic progressions.

4. For all positive integers and , there is a constant such that if and , , ..., satisfies , then of the numbers , , ..., are in arithmetic progression.

The constants are called van der Waerden numbers, and no formula for is known. van der Waerden's theorem is a corollary of Szemerédi's theorem.

Arithmetic Progression, Baudet's Conjecture, Szemerédi's Theorem, van der Waerden Number

This entry contributed by Kevin O'Bryant

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## References

de Bruijn, N. G. "Commentary." Unpublished manuscript, pp. 116-124, 1977. http://alexandria.tue.nl/repository/freearticles/598841.pdf.Guy, R. K. "Theorem of van der Waerden, Szemerédi's Theorem. Partitioning the Integers into Classes; at Least One Contains an A.P." §E10 in Unsolved Problems in Number Theory, 3rd ed. New York: Springer-Verlag, pp. 317-323, 2004.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. Wisk. 15, 212-216, 1927.van der Waerden, B. L. "How the Proof of Baudet's Conjecture Was Found." Studies in Pure Mathematics (Presented to Richard Rado). London: Academic Press, pp. 251-260, 1971.van der Waerden, B. L. "Wie der Beweis der Vermutung von Baudet gefunden wurde." Elem. Math. 53, 139-148, 1998.

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van der Waerden's Theorem

## Cite this as:

O'Bryant, Kevin. "van der Waerden's Theorem." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/vanderWaerdensTheorem.html