I am looking for the reference to the following (surely known) characterization of metric spaces that embed into $\mathbb R^n$:



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> **Theorem.** Let $n$ be positive integer number. A metric space $X$ is isometric to a subspace of the Euclidean space $\mathbb R^n$ if and only if every subspace $A\subseteq X$ of cardinality $|A|\le n+3$ is isometric to a subspace of $\mathbb R^n$.
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**Remark.** It is interesting to notice that $n+3$ in this theorem cannot be lowered to $n+2$. A counterexample in dimension 1 is the square
$$S=\{(1,1),(1,-1),(-1,1),(-1,-1)\}$$ in the plane $\mathbb R^2$ endowed with the $\ell\_1$-metric. Every 3-element subset of $S$ is isometric to a subspace of the real line but $S$ is not isometric to a subspace of a Euclidean space. Analogous examples can be constructed in any dimension.



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> **Example.** For every positive integer number $n$ there exists a metric space $X$ of cardinality $|X|=n+3$ such that every subspace $A\subseteq X$ of cardinality $|A|\le n+2$ is isometric to a subspace of $\mathbb R^n$ but $X$ is not isometric to a subspace of a Euclidean space.
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So, I am interested in finding references to the above Theorem and Example (or at least Theorem, which should be known). The proof of Theorem uses Kuratowski--Zorn Lemma and the rigidity of Euclidean spaces. The space $X$ in Example is a regular n-dimensional simplex with duplicated center. Can such an example have larger cardinality than $n+3$?



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> **Problem.** Is it true that a metric space $X$ of cardinality $|X|> n+3$ is isometric to a subspace of the Euclidean space $\mathbb R^n$ if every subset $A\subseteq X$ of cardinality $n+2$ is isometric to a subspace of $\mathbb R^n$.
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**Remark.** The Problem has affirmative solution for $n=1$.



A characterization of metric spaces, isometric to subspaces of Euclidean spaces
It is in


*Menger, K.*, [**Untersuchungen über allgemeine Metrik.**](https://doi.org/10.1007/BF01448840), Math. Ann. 100, 75-163 (1928). [EuDML:159284](https://eudml.org/doc/159284). [JFM 54.0622.02](https://zbmath.org/54.0622.02).


A modern exposition can be found in


*Bowers, John C.; Bowers, Philip L.*, [**A Menger redux: embedding metric spaces isometrically in Euclidean space**](https://doi.org/10.4169/amer.math.monthly.124.7.621), Am. Math. Mon. 124, No. 7, 621-636 (2017). [Zbl 1391.51007](https://zbmath.org/1391.51007).


