# equidistant set in $\mathbb{R}^n$

Let $X$ be a subset of $\mathbb{R}^n$ , such that given $x,y \in X$ we have $\lVert x-y\rVert = 1$. Prove that $|X| < n+2$.

This result is geometrically obvious, but how can I prove it for any $n$?

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WLOG one member of $X$ is $0$. If $x_1, \ldots, x_{n+1}$ are $n+1$ other members of $X$, consider the Gramian matrix $M$ with entries $x_i \cdot x_j$. Note that $M_{ii} = 1$ and $M_{ij} = 1/2$ for $i \ne j$. Show that the eigenvalues of $M$ are $1/2$ and $(n+1)/2$, and since $M$ is nonsingular, $x_1, \ldots, x_n$ must be linearly independent.