Erdős Problem #223

The problem

Let $d\geq 2$ and $n\geq 2$. Let $f_d(n)$ be maximal such that there exists some set of $n$ points $A\subseteq \mathbb{R}^d$, with diameter $1$, in which the distance 1 occurs between $f_d(n)$ many pairs of points in $A$. Estimate $f_d(n)$.

Statement quoted from erdosproblems.com/223, maintained by Thomas Bloom. Original source: [Er46b][Er57][Er75f,p.102]. Current status there: solved.

AI contributions

As recorded in Terence Tao's wiki AI contributions to Erdős problems (data as of 30 Jun 2026). 🟢 full, 🟡 partial, ⚪ unverified, 🔴 incorrect.

Caveats

The wiki's classifications are provisional and are not a benchmark. Many Erdős problems have never had a thorough literature review, so a result that looks new may later turn out to be known; some problems were also stated ambiguously or incorrectly. See the problem page for the full discussion.

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