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on Max to min ratios (AlphaEvolve problem 50)

The statement

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?

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.