Erdős Problem #864

Let \(A\subseteq \{1,\ldots,N\}\) be a set such that there exists at most one \(n\) with more than one solution to \(n=a+b\) (with \(a\leq b\in A\)). Estimate the maximal possible size of \(|A|\). In particular, is it true that \[|A| \leq (1+o(1))\frac{2}{\sqrt{3}} N^{1/2}?\] A problem of Erdős and Freud [ErFr91] [Er92c]. For the analogous question with \(n=a-b\) they prove that \(\lvert A\rvert\sim N^{1/2}\). This is a weaker form of Erdős Problem 840. The condition on \(A\) is `HasAtMostOneRepeatedSum`. We state the question as follows: for every \(\varepsilon > 0\) and all large \(N\), every such set \(A\) satisfies \(|A| \leq (1+\varepsilon)\frac{2}{\sqrt{3}} N^{1/2}\).