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What is the size of the largest such that there is a function such that
and
for all non-empty ?
Source: erdosproblems.com/319
No claim settles this problem.
Open on the site: the label is OPEN (2026-10-07; the page shows no last-edited stamp) and the site marks the problem as not resolvable by a finite computation. The standing derived from the claim pages is open, claim none: both claim pages are partial and pending, and a partial claim derives nothing. The site's commentary credits Adenwalla with the lower bound from Croot's Main Theorem (Acta Arith. 99 (2001); refereed), written out below and recorded on Adenwalla's claim page (2025); the credit is commentary on a problem the site labels OPEN, not an acceptance, so the bound is site-credited and pending. With the trivial upper bound it would give the order . The one claim on the site's proof-claim tab, an AI-assisted partial claim of 16 July 2026 that , is pending. No determination of the asymptotic, and no proof that , was found in the search whose scope the Current assessment records. This is a bounded negative finding, not a certificate of openness.