Max to min ratios

The problem

Let $n,d \geq 2$. Let $C(d,n)$ denote the largest quantity such that, given any $n$ distinct points $x_1,\dots,x_n$ in $\R^d$, the maximum distance $\max_{1 \leq i < j \leq n} \|x_i-x_j\|$ between the points is at least $C(d,n)$ times the minimum distance $\min_{1 \leq i < j \leq n} \|x_i-x_j\|$. Establish upper and lower bounds for $C(d,n)$. What are the configurations that attain the minimal ratio between the two distances?

Statement quoted from problem 50 of the AlphaEvolve repository of problems (Google DeepMind, CC BY 4.0).

What AI did

AlphaEvolve found a construction that improved on the best bound known at the time; the problem itself remains open. Status in the repository: New result. See Section 6.30 in "Mathematical Exploration and Discovery at Scale" paper. Records may have moved since; the repository's notes below list the latest bounds it knows of.

How it was checked

The results are explicit constructions or programs; the repository's notebooks include the verification code used to score them.

Limitations

An improved bound is progress, not a solution: the exact answer to the problem is still unknown. The authors report that results were better when experts set up and guided the search.

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