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Does there exist a maximal Sidon set $A\subset \{1,\ldots,N\}$ of size $O(N^{1/3})$?
A question of Erdős, Sárközy, and Sós [ESS94]. It is easy to prove that a maximal Sidon set in $\{1,\ldots,N\}$ has size $\gg N^{1/3}$. Ruzsa [Ru98b] constructed a maximal Sidon set of size $\ll (N\log N)^{1/3}$.