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Let be maximal such that in any finite set of size there exists a Sidon subset of size (i.e. the only solutions to in are the trivial ones). Determine the order of .
In particular, is it true that ?
Source: erdosproblems.com/530
No claim settles this problem.
Open, the site's label (OPEN). The order of is : the lower bound is the accepted partial claim Komlós, Sulyok and Szemerédi 1975, refereed, and the upper bound is the case with the Erdős--Turán bound for Sidon sets in an interval. Whether is open: the best lower constant is Bailleul and Riblet's [BaRi26], improving Abbott's [Ab90], and no source reaches .