High Quality Content by WIKIPEDIA articles! In number theory, Szemeredi's theorem refers to the proof of the Erdos–Turan conjecture. In 1936 Erdos and Turan conjectured 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(d,k)} of cardinality dN contains a length-k arithmetic 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 by an adaptation of the Hardy-Littlewood circle method. The case k = 4 was established in 1969 by Endre Szemeredi by a combinatorial method. Using an approach similar to the one he used for the case k = 3, Roth gave a second proof for this in 1972.

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