OPEN
This is open, and cannot be resolved with a finite computation.
Let $N\geq 1$ and $A\subset \{1,\ldots,N\}$ be a Sidon set. Is it true that, for any $\epsilon>0$, there exist $M$ and $B\subset \{N+1,\ldots,M\}$ (which may depend on $N,A,\epsilon$) such that $A\cup B\subset \{1,\ldots,M\}$ is a Sidon set of size at least $(1-\epsilon)M^{1/2}$?
See also
[329] and
[707] (indeed a positive solution to
[707] implies a positive solution to this problem, which in turn implies a positive solution to
[329]).
This is discussed in problem C9 of Guy's collection
[Gu04].
Additional thanks to: Gusarich and Desmond Weisenberg
