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Arithmetic Progressions Contained in Sequences with Bounded Gaps
Canadian mathematical bulletin, Tome 23 (1980) no. 4, pp. 491-493

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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. 
Nathanson, Melvyn B. Arithmetic Progressions Contained in Sequences with Bounded Gaps. Canadian mathematical bulletin, Tome 23 (1980) no. 4, pp. 491-493. doi: 10.4153/CMB-1980-074-x
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[1] 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

[2] 2. Rabung, J. R., On applications of van der Waerden's theorem, Math. Mag. 48 (1975), 142-148. Google Scholar

[3] 3. Szemerédi, E., On sets of integers containing no k elements in arithmetic progression, Acta Arithmetica 27 (1975), 199-245. Google Scholar

[4] 4. van der Waerden, B. L., Beweis einer Baudef′schen Vermutung, Nieuw Arch. Wiskunde 15 (1927), 212-216. Google Scholar

[5] 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

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