The one-exception Sidon problem

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established_bounds: The published leading bounds and the reflected construction behind the lower bound; the target remains unresolved.

source_notes/: Paper summaries and source comparisons used in the research on Problem 864.

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Let $M(N)$ be the largest size of $A\subseteq\{1,\ldots,N\}$ for which at most one integer has two or more unordered representations $a+b$, including diagonal pairs. The target is

$$M(N)\le\left(\frac2{\sqrt3}+o(1)\right)\sqrt N.$$

The published bounds are

$$\left(\frac2{\sqrt3}+o(1)\right)\sqrt N \le M(N) \le(2+o(1))\sqrt N.$$

The established bounds page gives the reflected construction behind the lower bound. The leading constant is undetermined.