A counterexample to Kusner's conjecture on equilateral sets

Abstract: We disprove Kusner's 1983 conjecture that every equilateral set in $\ell_p<sup>n$ with $2&lt;p&lt;\infty$ 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 $$e(\ell_5^{56})\ge58\&gt;57$$. This is the first equilateral set of more than $n+1$ points in $\ell_p<sup>n$ 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.

• The paper disproves Kusner’s conjecture by rigorously certifying an equilateral set of 58 points in ℓ₅⁵⁶, exceeding the conjectured maximum of n+1 = 57.
• The authors use a computer-generated 58×56 integer certificate, exact rational arithmetic, and a Newton–Kantorovich contraction argument to prove existence, uniqueness, and nondegeneracy.
• The configuration persists for exponents in an open interval around p = 5, placing the first failure threshold in the range 4 ≤ p₀ < 5 while leaving its exact value and higher-dimensional behavior open.

The conjecture and its status prior to this work

Kusner conjectured in 1983 that every equilateral set in $\ell_p^n$, for $2<p<\infty$, contains at most $n+1$ points, matching the Euclidean case $p=2$. The conjecture was supported by a general-position heuristic: each additional point of an equilateral set must lie on the smooth $\ell_p$-spheres about all preceding points, and each such sphere should cut the dimension by one. Prior confirmations covered all $p$ sufficiently close to $2$ (Smyth), $p=4$ with an $n$-dependent interval about $4$ (Swanepoel), and — in work of Ge, Xu and Zhou — the entire range $2<p<\infty$0. For $2<p<\infty$1, only upper bounds were available, the sharpest at $2<p<\infty$2 being $2<p<\infty$3 (Alon–Pudlák). Notably, no equilateral set of more than $2<p<\infty$4 points was known in $2<p<\infty$5 for any finite $2<p<\infty$6, whereas for $2<p<\infty$7 Swanepoel had shown the conjecture false with sets of $2<p<\infty$8 points.

The main result

The paper establishes that $2<p<\infty$9: there exist $n+1$0 points in $n+1$1 with all pairwise $n+1$2 distances equal. This is the first equilateral set exceeding $n+1$3 points in $n+1$4 for any finite $n+1$5, and it disproves Kusner's conjecture in its original range. Moreover, by the implicit function theorem, the configuration persists on an open interval of exponents around $n+1$6, so the threshold

$n+1$7

satisfies $n+1$8; combined with the confirmed range $n+1$9, one obtains $p=2$0. The lower bound $p=2$1 is the only part of this conclusion that depends on Ge–Xu–Zhou; the counterexample at $p=2$2 is independent of their work.

The exponent $p=2$3 is significant: it is the first integer beyond the verified range, and it lies in one of the intervals $p=2$4 that Ge, Xu and Zhou identify as "genuinely harder" — the standard algebraic techniques are naturally adapted to even exponents, and only the almost-linear $p=2$5 upper bounds apply at odd exponents.

The construction and its verification

The configuration is specified by a computer-searched certificate: a $p=2$6 integer matrix $p=2$7 (entries below $p=2$8) giving dyadic centre coordinates, a dyadic centre value $p=2$9 for the common fifth-power distance, a set of $\ell_p$0 selected coordinate positions out of $\ell_p$1, and a rational $\ell_p$2 preconditioner. The remaining $\ell_p$3 coordinates are frozen at centre values. Writing $\ell_p$4 for the vector of selected coordinates plus the unknown common distance $\ell_p$5, one must solve $\ell_p$6 where

$\ell_p$7

a system of $\ell_p$8 equations in $\ell_p$9 unknowns. A key structural observation is that the minimum coordinate gap at the centre exceeds $p$0 with box radius $p$1, so every coordinate difference keeps a fixed nonzero sign on the box. Since the exponent $p$2 is odd, $p$3 is therefore a polynomial with rational coefficients on the entire box, and the verification reduces to exact integer arithmetic.

Existence and uniqueness follow from a Newton–Kantorovich contraction lemma: with the preconditioner $p$4, the certified quantities are $p$5 and $p$6 (exact values $p$7, $p$8), so the map $p$9 is a contraction on the box mapping it into itself. The zero $2$0 is unique, lies in the interior of the box, has invertible Jacobian, and is real algebraic. The Jacobian deviation bound exploits monotonicity of $2$1 on $2$2 together with the sign constancy of coordinate differences.

The verification is fully rigorous: all quantities are rationals with denominators dividing $2$3, and the six certificate claims are finite integer computations performed in exact arithmetic from the archived data.

Persistence in the exponent

Because all coordinate differences are bounded away from zero on the box, $2$4 extends analytically in the exponent $2$5 via $2$6 with fixed signs. Since $2$7 is a nondegenerate zero (invertible Jacobian), the implicit function theorem yields a branch of equilateral configurations of $2$8 points in $2$9 for all $p=4$0 in some interval around $p=4$1.

The paper is explicit that this argument gives no quantitative control of the interval: the branch may leave the box or cease to exist well inside $p=4$2, and the certificate does not transfer to non-integer $p=4$3, since it is the oddness of the exponent that makes the system polynomial with dyadic coefficients.

Limitations and open questions

The result settles the conjecture only at the single instance $p=4$4 (and nearby exponents). Whether $p=4$5 for infinitely many $p=4$6, and whether $p=4$7 is bounded if so, remain open; even linearity of $p=4$8 in $p=4$9 is unknown, with the best upper bound of order $n$0. In the other direction, known results imply $n$1 whenever $n$2, so if $n$3 — the second open question posed — any failure approaching $n$4 from above would require witnessing dimensions growing without bound. The paper also notes that no symmetry or compressed description of the configuration is known.

Conclusion

The paper refutes Kusner's 1983 conjecture by certifying, via an exact-arithmetic Newton–Kantorovich argument, a unique equilateral set of $n$5 points in $n$6, the first such set exceeding $n$7 for any finite $n$8. The configuration persists on an open interval of exponents about $n$9, placing the failure threshold $4$0 in $4$1 and sharply focusing attention on the behaviour of equilateral numbers immediately above the exponent $4$2.