- Szemerédi's theorem
In

number theory **Szemerédi's theorem**refers to the proof of the**Erdős–Turán conjecture**. In 1936 Erdős and Turan conjecturedcitation|authorlink1=Paul Erdős|first1=Paul|last1=Erdős|authorlink2=Paul Turán|first2=Paul|last2=Turán|title=On some sequences of integers|journal=Journal of the London Mathematical Society |volume=11|year=1936|pages=261–264.] for every value "d" called density 0 < "d" <1 and every integer "k" there is a number "N"("d","k") such that every subset A of {1,...,"N"} ofcardinality "dN" contains a length-karithmetic progression , provided "N" > "N"("d","k").This generalizes the statement of

van der Waerden's theorem .The cases "k=1" and "k=2" are trivial. The case "k" = 3 was established in 1956 by

Klaus Roth [*citation|authorlink=Klaus Friedrich Roth|first=Klaus Friedrich|last=Roth|title=On certain sets of integers, I|journal=*] by an adaptation of theJournal of the London Mathematical Society |volume=28|pages=104–109|year=1953|id=Zbl 0050.04002, MathSciNet|id=0051853.Hardy-Littlewood circle method . The case "k" = 4 was established in 1969 byEndre Szemerédi [*citation|authorlink=Endre Szemerédi|first=Endre|last=Szemerédi|title=On sets of integers containing no four elements in arithmetic progression|journal=Acta Math. Acad. Sci. Hung.|volume=20|pages=89–104|year=1969|id=Zbl 0175.04301, MathSciNet|id=0245555|doi=10.1007/BF01894569.*] by a combinatorial method. Using an approach similar to the one he used for the case "k" = 3, Roth [*citation|authorlink=Klaus Friedrich Roth|first=Klaus Friedrich|last=Roth|title=Irregularities of sequences relative to arithmetic progressions, IV|journal=Periodica Math. Hungar.|volume=2|pages=301–326|year=1972|id=MathSciNet|id=0369311|doi=10.1007/BF02018670.*] gave a second proof for this in 1972.Finally, the case of general "k" was settled in 1975, also by Szemerédi, [

*citation|authorlink=Endre Szemerédi|first=Endre|last=Szemerédi|title=On sets of integers containing no "k" elements in arithmetic progression|journal=Acta Arithmetica|volume=27|pages=199–245|year=1975|id=Zbl 0303.10056, MathSciNet|id=0369312.*] by an ingenious and complicated extension of the previous combinatorial argument ("a masterpiece of combinatorial reasoning", R. L. Graham). Important alternative proofs of this theorem were later found byHillel Furstenberg [*citation|authorlink=Hillel Furstenberg|first=Hillel|last=Furstenberg|title=Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions|journal=J. d’Analyse Math.|volume=31|pages=204–256|year=1977|id=MathSciNet|id=0498471.*] in 1977, usingergodic theory , and byTimothy Gowers citation|authorlink=Timothy Gowers|first=Timothy|last=Gowers|title=A new proof of Szemerédi's theorem|journal=Geom. Funct. Anal.|volume=11|issue=3|pages=465–588|year=2001|id=MathSciNet|id=1844079.] in 2001, using bothFourier analysis and combinatorics.Let "k" be a positive integer and let 0 < δ ≤ 1/2. A

finitary version of the theorem states that there exists a positive integer:"N" = "N"("k", δ)

such that every subset of {1, 2, ..., "N"} of size at least δ"N" contains an arithmetic progression of length "k".The best-known bounds for "N"("k", δ) are

:$C^\{log(1/delta)^\{k-1\; leq\; N(k,delta)\; leq\; 2^\{2^\{delta^\{-2^\{2^\{k+9\}$

with "C" > 0. The lower bound is due to

Behrend [*citation|authorlink=Felix A. Behrend|first=Felix A.|last=Behrend|title=On the sets of integers which contain no three in arithmetic progression|journal=*] (for "k" = 3) and Rankin, [Proceedings of the National Academy of Sciences |volume=23|pages=331–332|year=1946|id=Zbl 0060.10302.*citation|authorlink=Robert A. Rankin|first=Robert A.|last=Rankin|title=Sets of integers containing not more than a given number of terms in arithmetical progression|journal=Proc. Roy. Soc. Edinburgh Sect. A|volume=65|pages=332–344|year=1962|id=Zbl 0104.03705, MathSciNet|id=0142526.*] and the upper bound is due to Gowers. When "k" = 3 better upper bounds are known; the current record is:$N(3,delta)\; leq\; C^\{delta^\{-2\}log(1/delta)\},$

due to

Bourgain . [*citation|authorlink=Jean Bourgain|first=Jean|last=Bourgain|title=On triples in arithmetic progression|journal=Geom. Func. Anal.|volume=9|year=1999|pages=968–984|id=MathSciNet|id=1726234|doi=10.1007/s000390050105.*]**ee also***

Problems involving arithmetic progressions

*Ergodic Ramsey theory

*Arithmetic combinatorics **References****External links*** [

*http://planetmath.org/encyclopedia/SzemeredisTheorem.html PlanetMath source for initial version of this page*]

* [*http://www.math.ucla.edu/~tao/whatsnew.html Announcement by Ben Green and Terence Tao*] - the preprint is available at [*http://front.math.ucdavis.edu/math.NT/0404188 math.NT/0404188*]

* [*http://in-theory.blogspot.com/2006/06/szemeredis-theorem.html Discussion of Szemerédi's theorem (part 1 of 5)*]

*Ben Green and Terence Tao: [*http://www.scholarpedia.org/article/Szemeredi%27s_Theorem Szemerédi's theorem*] onScholarpedia .

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