Claims

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1983_06_01_bannai_bannai_stanton: A set in $\mathbb R^n$ whose points determine only two distinct distances has at most $\binom{n+2}{2}$ points; more generally an $s$-distance set has at most $\binom{n+s}{s}$ points.

1997_02_01_lisonek: Correct, but answers the site's wording (the exact maximum, for n at most 8), not the corrected Statement (the asymptotic behavior of M_2(n)), so it does not count toward the problem's standing. Lisoněk (1997) determines the largest two-distance sets in R^n for every n at most 8: 3, 5, 6, 10, 16, 27, 29 and 45 points, the last attaining the Bannai–Bannai–Stanton bound.

2019_12_17_petrov_pohoata: A new proof, through the inertia of a polynomial matrix, that an $s$-distance set in $\mathbb R^n$ has at most $\binom{n+s}{s}$ points, so a two-distance set has at most $\binom{n+2}{2}$.

2025_08_09_alweiss: The construction the site's curator credits to Ryan Alweiss: the points e_i + e_j lie on a hyperplane and give a two-distance set of binom(n+1,2) points in R^n, the lower bound; accepted on the curator's credit.

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