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Arithmetic Progressions Contained in Sequences with Bounded Gaps

Published online by Cambridge University Press:  20 November 2018

Melvyn B. Nathanson
Melvyn B. Nathanson*
Affiliation:
Department of MathematicsSouthern Illinois University, Carbondale Illinois 62901
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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.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 23 , Issue 4 , 01 December 1980 , pp. 491 - 493
Copyright
Copyright © Canadian Mathematical Society 1980

References

1. 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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3. Szemerédi, E., On sets of integers containing no k elements in arithmetic progression, Acta Arithmetica 27 (1975), 199-245.Google Scholar
4. van der Waerden, B. L., Beweis einer Baudef′schen Vermutung, Nieuw Arch. Wiskunde 15 (1927), 212-216.Google Scholar
5. 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