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Let be the sequence defined by and , and for define as the least positive integer such that there is no three-term arithmetic progression in .
Can the be explicitly determined? How fast do they grow?
Source: erdosproblems.com/271
No claim settles this problem.
Open, the site's label (OPEN; page last edited 20 January 2026). The site's commentary credits Odlyzko and Stanley [OdSt78] with explicit descriptions of and for every , which answer both questions for those ; the memorandum states them without proof, and they are recorded as a claimed partial claim on its claim page (Odlyzko and Stanley, 1978). Rolnick proved them for , with growth of order , in a refereed paper, recorded as an accepted partial claim on his claim page (Rolnick, 2014). Moy's bound [Mo11] holds for every and fixes no sequence's growth, so it settles no instance and has no claim page. For every other both questions are open.