---

Claim. The function $f(N)$ of Problem 819, the largest number of sums in $[1,N]$ of a subset of $\{1,\ldots,N\}$ with $\lfloor N^{1/2}\rfloor$ elements, satisfies

$$\liminf_{N\to\infty}\frac{f(N)}{N}\ge\frac{16\sqrt2-17}{12}\approx0.4690,$$

the note's Theorem 3, against the lower bound $3/8=0.375$ of Erdős and Freud recorded on the problem page and the trivial upper bound $f(N)\le N/2+O(N^{1/2})$. The construction, as the note describes it: for $N=4q^2$ with $q$ large, take an asymptotically maximum Sidon set $S\subset[0,M]$ of size $q$, $M=(1+o(1))q^2$, obtained by thinning a Bose--Chowla set, fix $u\in(0,1/4)$ and $\eta\in(0,u/2)$, and set $A=(a+S)\cup(N/2+\lfloor uN/2\rfloor-b-S)$ with $a$ and $b$ independent and uniform in $\{1,\ldots,\lfloor\eta N/2\rfloor\}$, so that both copies are shifted at random (the note says the construction and proof strategy are inspired by Pikhurko's Lemma 12); the overlap between the Sidon set and its reflection is controlled by Pikhurko's uniformity lemma for asymptotically maximum Sidon sets (Lemma 10 of the paper on the card pikhurko_2006_dense_edge_magic_graphs_thin_additive), the expected number of sums in $[1,N]$ is at least $(N/2)F(u)-O(\eta N)-o(N)$ with $F(u)=1-2u^2-(1-2u)^3/12$ (Theorem 20; the proof of Theorem 3 lets $\eta\to0$), maximized at $u^*=3/2-\sqrt2$, where $F(u^*)=(16\sqrt2-17)/6\approx0.938$, so the constant is $F(u^*)/2$ (Corollary 25); an interpolation argument carries the bound from the subsequence $N=4q^2$ to all large $N$. The note is dated 15 May 2026 and hosted on the author's personal site; its acknowledgments say that the proof was found with the assistance of OpenAI's GPT-5.5 together with the Rethlas open-source agentic mathematics-research pipeline and that the author verified every argument and takes responsibility for the paper. The argument is not checked in this corpus.

Covers. The lower bound alone: $\liminf f(N)/N\ge(16\sqrt2-17)/12$, above the refereed $\tfrac38$ of Erdős and Freud. The claim does not determine the asymptotic size of $f(N)/N$, which the problem asks for (its order $\Theta(N)$ being known), and says nothing about the upper bound $1/2+o(1)$ or about the quasi-Sidon question of Problem 840 to which Erdős and Freud tie any improvement of that upper bound.

Depends on. No page of this wiki; Pikhurko's lemma and the Bose--Chowla construction are literature inputs.

Standing. Claimed. The note was announced in the site's discussion thread on 2026-05-15 and is not on the proof-claims tab, which is empty; the site labels the problem OPEN (as of 2026-10-07), and its commentary, which gives the bounds $3/8$ and $1/2$, does not mention it. A second thread commenter reported on 2026-05-17 having run a standard check, linking a chat transcript, which found no issue; that is not a review. The note is not on a preprint server, has no journal record, and has no formalization.