OPEN - $1000
Let $h(N)$ be the maximum size of a Sidon set in $\{1,\ldots,N\}$. Is it true that, for every $\epsilon>0$,
\[h(N) = N^{1/2}+O_\epsilon(N^\epsilon)?\]
A problem of Erdős and Turán. It may even be true that $h(N)=N^{1/2}+O(1)$, but Erdős remarks this is perhaps too optimistic. Erdős and Turán
[ErTu41] proved an upper bound of $N^{1/2}+O(N^{1/4})$, with an alternative proof by Lindström
[Li69]. Both proofs in fact give
\[h(N) \leq N^{1/2}+N^{1/4}+1.\]
Balogh, Füredi, and Roy
[BFR21] improved the bound in the error term to $0.998N^{1/4}$, which has been further optimised by O'Bryant
[OB22] to yield
\[h(N)\leq N^{1/2}+0.99703N^{1/4}\]
for sufficiently large $N$.
See also [241] and [840].
