Problem Statement
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 $(1-\epsilon)M^{1/2}$?
Categories:
Number Theory Sidon Sets Additive Combinatorics
Progress
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].
Source: erdosproblems.com/44 | Last verified: January 13, 2026