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Szemerédi's Theorem

Szemerédi's theorem states that every sequence of integers that has positive upper Banach density contains arbitrarily long arithmetic progressions.

A corollary states that, for any positive integer k and positive real number delta, there exists a threshold number n(k,delta) such that for n>=n(k,delta) every subset of {1,2,...,n} with cardinal number larger than deltan contains a k-term arithmetic progression. van der Waerden's Theorem follows immediately by setting delta=n/r. The best bounds for van der Waerden numbers are derived from bounds for n(k,delta) in Szemerédi's theorem.

Szemerédi's theorem was conjectured by Erdős and Turán (1936). Roth (1953) proved the case k=3, and was mentioned in his Fields Medal citation. Szemerédi (1969) proved the case k=4, and the general theorem in 1975 as a consequence of Szemerédi's regularity lemma (Szemerédi 1975a), for which he collected a $1000 prize from Erdos. Fürstenberg and Katznelson (1979) proved Szemerédi's theorem using ergodic theory. Gowers (1998ab) subsequently gave a new proof, with a better bound on n(k,r), for the case k=4 (mentioned in his Fields Medal citation; Lepowsky et al. 1999).

See also

Arithmetic Progression, Banach Density, Erdős-Turán Conjecture, Szemerédi's Regularity Lemma, van der Waerden Number, van der Waerden's Theorem

This entry contributed by Kevin O'Bryant

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References

Erdős, P. and Turán, P. "On Some Sequences of Integers." J. London Math. Soc. 11, 261-264, 1936.Fürstenberg, H. "Ergodic Behavior of Diagonal Measures and a Theorem of Szemerédi on Arithmetic Progressions." J. Analyse Math. 31, 204-256, 1977.Fürstenberg, H. and Katznelson, Y. "An Ergodic Szemerédi Theorem for Commuting Transformations." J. Analyse Math. 34, 275-291, 1979.Fürstenberg, H. and Weiss, B. "A Mean Ergodic Theorem for 1/Nsum_(n=1)^(N)f(T^nx)g(T^(n^2)x)." In Convergence in Ergodic Theory and Probability (Columbus OH 1993). Berlin: de Gruyter, pp. 193-227, 1996.Fürstenberg, H.; Katznelson, Y.; and Ornstein, D. "The Ergodic-Theoretical Proof of Szemerédi's Theorem." Bull. Amer. Math. Soc. 7, 527-552, 1982.Gowers, W. T. "Fourier Analysis and Szemerédi's Theorem." In Proceedings of the International Congress of Mathematicians, Vol. I (Berlin, 1998). Doc. Math., Extra Vol. I, 617-629, 1998a.Gowers, W. T. "A New Proof of Szemerédi's Theorem for Arithmetic Progressions of Length Four." Geom. Funct. Anal. 8, pp. 529-551, 1998b.Gowers, W. T. "A New Proof of Szemerédi's Theorem." Geom. Funct. Anal. 11, 465-588, 2001.Graham, R. L.; Rothschild, B. L.; and Spencer, J. H. Ramsey Theory, 2nd ed. New York: Wiley, 1990.Green, B. and Tao, T. "The Primes Contain Arbitrarily Long Arithmetic Progressions." Preprint. 8 Apr 2004. http://arxiv.org/abs/math.NT/0404188.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, 2nd ed. New York: Springer-Verlag, pp. 204-209, 1994.Lepowsky, J.; Lindenstrauss, J.; Manin, Y.; and Milnor, J. "The Mathematical Work of the 1998 Fields Medalists." Not. Amer. Math. Soc. 46, 17-26, 1999.Roth, K. "Sur quelques ensembles d'entiers." Comptes Rendus Acad. Sci. Paris 234, 388-390, 1952.Roth, K. F. "On Certain Sets of Integers." J. London Math. Soc. 28, 104-109, 1953.Szemerédi, E. "On Sets of Integers Containing No Four Elements in Arithmetic Progression." Acta Math. Acad. Sci. Hungar. 20, 89-104, 1969.Szemerédi, E. "On Sets of Integers Containing No k Elements in Arithmetic Progression." Acta Arith. 27, 199-245, 1975a.Szemerédi, E. "On Sets of Integers Containing No k Elements in Arithmetic Progression." In Proceedings of the International Congress of Mathematicians, Volume 2, Held in Vancouver, B.C., August 21-29, 1974. Montreal, Quebec: Canad. Math. Congress, pp. 503-505, 1975b.

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Szemerédi's Theorem

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

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