Problem 754

Let $f(n)$ be maximal such that there exists a set $A$ of $n$ points in $\mathbb{R}^4$ in which every $x\in A$ has at least $f(n)$ points in $A$ equidistant from $x$ .

Is it true that $f(n)\leq \frac{n}{2}+O(1)$ ?

Source: erdosproblems.com/754

Geometry
Distances

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SolvedProved

An accepted solution exists. The statement is true.

Proved. The site marks the problem proved and credits Swanepoel's bound on favorite distances in four dimensions (label PROVED in the site's export of 2026-09-04; the community database records proved (Lean)); see the claim page (Swanepoel, 2011).

Problem 754

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