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Arithmetic Progressions Contained in Sequences with Bounded Gaps
Published online by Cambridge University Press: 20 November 2018
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Van der Waerden [1, 4, 5] proved that if the nonnegative integers are partitioned into a finite number of sets, then at least one set in the partition contains arbitrarily long finite arithmetic progressions. This is equivalent to the result that a strictly increasing sequence of integers with bounded gaps contains arbitrarily long finite arithmetic progressions. Szemerèdi [3] proved the much deeper result that a sequence of integers of positive density contains arbitrarily long finite arithmetic progressions. The purpose of this note is a quantitative comparison of van der Waerden's theorem and sequences with bounded gaps.
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- Copyright © Canadian Mathematical Society 1980
References
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Graham, R. L. and Rothschild, B. L., A short proof of van der Waerden′s theorem on arithmetic progressions, Proc. Amer. Math. Soc. 42 (1974), 385-386.Google Scholar
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Szemerédi, E., On sets of integers containing no k elements in arithmetic progression, Acta Arithmetica
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van der Waerden, B. L., Beweis einer Baudef′schen Vermutung, Nieuw Arch. Wiskunde
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van der Waerden, B. L., How the proof of Baudet′s conjecture was found, Studies in Pure Mathematics, L. Mirsky, éd., Academic Press, New York, 1971, 251-260.Google Scholar
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