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Claim. Let $A\subseteq\mathbb R^n$ be a finite set whose nonzero pairwise distances take exactly $s$ values. Then $|A|\le\binom{n+s}{s}$. For $s=2$ this bounds the sets of Problem 502 by $\binom{n+2}{2}$, and since an infinite two-distance set would contain finite two-distance subsets of every size, no infinite such set exists.

Covers. The part upper_bound of the corrected Statement: the bound $|A|\le\binom{n+2}{2}$ on every two-distance set in $\mathbb R^n$. With the lower construction of $\binom{n+1}{2}$ points, which settles the other part, it gives the asymptotic behavior $n^2/2+O(n)$ of the largest size, which the problem asks for.

The argument. The paper is the second part of the authors' work on $s$-distance sets and proves the bound through the linear independence of a family of polynomials attached to the points. Its theorem is restated and reproved, by a different method, in Petrov and Pohoata's note, which has its own claim page in this folder and whose [[../library/distance_problems/petrov_2021_remark_sets_few_distances/theorem_1_1|Theorem 1.1 page]] gives the complete proof of the same bound.

Acceptance. The paper is refereed: E. Bannai, E. Bannai and D. Stanton, An upper bound for the cardinality of an $s$-distance subset in real Euclidean space, II, Combinatorica 3 (1983), no. 2, 147–152. The curator of erdosproblems.com, Thomas Bloom, marks the problem solved and credits the upper bound to this paper.