Optimal dimension reduction bound for Euclidean point sets

Let $f(n,d,\varepsilon)$ denote the smallest $m$ such that every $n$-point subset of $\ell_2^d$ can be embedded into $\ell_2^m$ with distortion at most $1+\varepsilon$. Dimension-reduction conjecture. For all $n,d>1$ and $0<\varepsilon<1$,

$$f(n,d,\varepsilon)=\Theta\left(\min\left\{n,d,\varepsilon^{-2}\lg(2+\varepsilon^2n)\right\}\right).$$

This conjecture predicts the optimal Euclidean target dimension across the full range of distortion parameters, including the regime where $\varepsilon$ approaches $1/\sqrt{\min\{n,d\}}$; the paper proves a matching lower bound up to a logarithmic factor in a nearly full range of $\varepsilon$, while the stated formula remains the conjectural optimal form.