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Claim. For every $n\geq3$ there is a two-distance set of $\binom{n+1}{2}$ points in $\mathbb R^n$, so the largest two-distance set of Problem 502 has at least $\binom{n+1}{2}$ points. The construction, which the site's curator records in the commentary of Problems 502 and 503 and credits to Ryan Alweiss, takes the vectors $e_i+e_j$ for distinct coordinate vectors $e_i,e_j$ of $\mathbb R^{n+1}$. They lie on the hyperplane on which the coordinates sum to $2$, a copy of $\mathbb R^n$. Two of them are at distance $\sqrt2$ when their index pairs share one index and at distance $2$ when the pairs are disjoint, and for $n\geq3$ both cases occur. They are the midpoints of the edges of the regular $n$-simplex with vertices $2e_1,\ldots,2e_{n+1}$. The commentary of Problem 502 credits Shengtong Zhang with the earlier lower bound $\binom n2$, from the points of $\mathbb R^n$ with exactly two coordinates equal to $1$ and all others $0$, and obtains $\binom{n+1}{2}$ from the observation that these points lie on a hyperplane, citing Alweiss's construction.

Covers. The part lower_bound of the corrected Statement: the bound $\binom{n+1}{2}$ for $n\geq3$. With the upper bound $\binom{n+2}{2}$, which settles the other part, it gives the asymptotic behavior $n^2/2+O(n)$ of the largest size, which the problem asks for. Not covered: the exact maximum, which exceeds $\binom{n+1}{2}$ in some dimensions and is a variant under the problem page's Formulation.

Depends on. The library's [[../library/distance_problems/ge_2026_two_distance_set_277_points_23_dimensions/intro_midpoint_construction|introductory midpoint construction]], which states the construction from Ge, Koolen and Munemasa's introduction and verifies its two distances.

Acceptance. Reviewed: the curator of erdosproblems.com, T. F. Bloom, records the construction and credits it to Alweiss in the commentary of Problem 503, and the commentary of Problem 502 (page last edited 29 January 2026) cites it for the lower bound $\binom{n+1}{2}$ and labels the problem solved. Not refereed: no paper by Alweiss states the construction; Ge, Koolen and Munemasa's refereed paper states it in its introduction as the standard lower construction.

Postings and dating. The commentary carries no date. The site's history views of Problems 502 and 503 show the construction already present in their earliest listed revision, of 20 October 2025. The first dated reference to it is DesmondWeisenberg's post of 9 August 2025 in the discussion thread of Problem 503, which improves "Alweiss' construction" by one point for that problem; that date names this page.