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Published May 2021 | Submitted
Journal Article Open

Sets without k-term progressions can have many shorter progressions

Abstract

Let f_(s, k)(n) be the maximum possible number of s‐term arithmetic progressions in a set of n integers which contains no k‐term arithmetic progression. For all fixed integers k > s ≥ 3, we prove that f_(s, k)(n) = n^(2 − o(1)), which answers an old question of Erdős. In fact, we prove upper and lower bounds for f_(s, k)(n) which show that its growth is closely related to the bounds in Szemerédi's theorem.

Additional Information

© 2020 Wiley Periodicals LLC. Issue Online: 10 March 2021; Version of Record online: 15 December 2020; Manuscript accepted: 27 July 2020; Manuscript received: 10 April 2020. Funding Information: NSF Grant Number: DMS‐1855635.

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Additional details

Created:
August 22, 2023
Modified:
October 18, 2023