Logan R. Chalmers

A counterexample to Kusner's conjecture on equilateral sets

We disprove Kusner's 1983 conjecture that every equilateral set in $\ell_p^n$ with $2<p<\infty$ has at most $n+1$ points: there exist $58$ points in $\mathbb{R}^{56}$ whose pairwise $\ell_5$ distances are all equal, so the maximum equilateral-set size satisfies $e(\ell_5^{56})\ge58>57$. This is the first equilateral set of more than $n+1$ points in $\ell_p^n$ for any finite $p\ge2$. The construction persists on an open interval of exponents around $5$; since Ge, Xu and Zhou recently proved the conjecture for $2\le p\le4$, the infimum of exponents at which it fails lies in $[4,5)$. The configuration is the unique solution of an explicit polynomial system with rational coefficients in a rational box, established in exact arithmetic.