The exact asymptotic constant in the metric dimension of Jaccard space

Let $X$ be a finite set with $\mathrm{\mid }X\mathrm{\mid }=n$ and let $\mathrm{J}\mathrm{a}\mathrm{c}(a,b)=\mathrm{\mid }a\mathrm{△}b\mathrm{\mid }\mathrm{/}\mathrm{\mid }a\cup b\mathrm{\mid }$ be the Jaccard distance on the power set $2^X$. Lladser and Paradise recently proved that the metric dimension of $(2^X,\mathrm{J}\mathrm{a}\mathrm{c})$ is $\mathrm{\Theta }(n\mathrm{/}\ln n)$, with the constant left open; their bounds are $(\ln 2)n\mathrm{/}\ln n\lesssim \beta (2^X,\mathrm{J}\mathrm{a}\mathrm{c})\lesssim 2\ln (2e)n\mathrm{/}\ln n$. We determine the constant: $$\beta (2^X,\mathrm{J}\mathrm{a}\mathrm{c})=\frac{2n}{{\log }_{2}n}(1+o(1))=(2\ln 2)\frac{n}{\ln n}(1+o(1))\mathrm{.}$$ The proof identifies the problem, on each ``slice’’ of subsets of fixed cardinality, with the Erdős–Rényi coin-weighing problem for a spring scale (the problem of \emph{detecting matrices}). The lower bound is the Erdős–Rényi entropy argument applied to the middle slice; the upper bound follows from the explicit detecting families of Lindström and of Cantor and Mills, augmented by a single extra landmark that reveals cardinality.