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Claim. Let $M_2(n)$ be the largest size of a set in $\mathbb R^n$ whose points determine exactly two distinct distances, the quantity whose exact value the site's wording of Problem 502 asks for. Lisoněk determines it for every $n\leq8$:

$$M_2(1),\ldots,M_2(8)=3,\ 5,\ 6,\ 10,\ 16,\ 27,\ 29,\ 45.$$

The paper is P. Lisoněk, New maximal two-distance sets, J. Combin. Theory Ser. A 77 (1997), no. 2, 318–338, DOI 10.1006/jcta.1997.2749. Its new sets are, in $\mathbb R^7$, a set of $29=\binom82+1$ points, one more than the $\binom{n+1}{2}=28$ midpoints of the edges of a regular $7$-simplex, and, in $\mathbb R^8$, a set of $45=\binom{10}{2}$ points containing it, which attains the upper bound $\binom{n+2}{2}$ of Bannai, Bannai and Stanton. Ge, Koolen and Munemasa's introduction (card) credits Lisoněk with determining the sizes of the largest two-distance sets in dimensions $d\leq8$ and cites Lisoněk's Theorem 4.4 for the uniqueness, up to isometry and scaling, of the $29$-point set in $\mathbb R^7$; the sequence of maxima is recorded as A027627 in the OEIS with the paper as its reference. The paper is not held in the library.

Covers. The exact value of $M_2(n)$ for $n\leq8$. The claim's value is answered because the result determines, for these dimensions, what the site's wording asks to determine. Not covered: the asymptotic behavior of $M_2(n)$, which the corrected Statement asks for, and any $n\geq9$, where the bounds $\binom{n+1}{2}\leq M_2(n)\leq\binom{n+2}{2}$ on the problem page leave a gap, improved at one dimension by Ge, Koolen and Munemasa's $277$-point set in $\mathbb R^{23}$.

Standing. Rejected: the result determines the exact maximum for $n\leq8$, the site's wording's question; the corrected Statement asks for the asymptotic behavior, which finitely many values do not reach. The problem page records the exact maximum as a variant under its Formulation and credits the values there.

Depends on. No page of this wiki.

Acceptance. The refereed evidence is the journal publication cited above, in the Journal of Combinatorial Theory, Series A. The site's remarks do not mention the result, so no reviewed evidence is listed. The record dates the issue to February 1997 with no finer date, so the page carries the first day of that month.