Erdős's diameter conjecture for separated distances fails in high dimensions

Abstract: Erdős asked whether every $n$-point set in Euclidean space whose $\binom{n}{2}$ pairwise distances are mutually at least $1$ apart must have diameter at least $(1+o(1))n^2$. We disprove this statement by constructing for every prime power $q$ a set $\mathcal X_q\subset \mathbb R^{q^2+q}$ of $n=q+1$ points such that all pairwise distances in $\mathcal X_q$ are mutually at least $1$ apart, while $$\operatorname{diam}(\mathcal X_q)\le\Bigl(1-\frac{1}{π^2}+o(1)\Bigr)n^2.$$ The proof is fully formalized in Lean 4.

• The paper constructs explicit high-dimensional n-point sets with 1-separated distances, showing that the conjectured quadratic diameter lower bound is violated.
• The construction employs Singer sets and scaling methods to precisely control distance separation and achieve an upper bound of roughly (1-1/π²)n².
• The findings challenge established geometric intuition and open new questions regarding optimal bounds in fixed low-dimensional Euclidean spaces.

Failure of Erdős's Diameter Conjecture for Separated Distances in High Dimensions

Introduction and Background

The paper "Erdős's diameter conjecture for separated distances fails in high dimensions" (2604.15305) addresses a longstanding problem in combinatorial geometry, originally posed by Erdős. The conjecture concerns the minimal diameter of $n$-point sets in Euclidean space when the multiset of all pairwise distances is $1$-separated, that is, any two distinct distances differ by at least $1$. While Erdős provided a proof for the case $d=1$, he conjectured that for any $n$ and in all dimensions, such point sets must have diameter at least $(1+o(1))n^2$—a dimension-independent quadratic lower bound. The analysis of this problem connects to the broader study of constraints imposed by prescribed separation (or sparsity) in the distance multiset on extremal properties of finite point configurations.

Main Result and Construction

The central result establishes that Erdős’s conjectured lower bound is asymptotically false in high-dimensional Euclidean spaces. For infinitely many $n$, the paper constructs explicit $n$-point sets $\mathcal X_n \subset \mathbb R^{n^2-n}$ with all pairwise distances separated by at least $1$, but with diameter

$1$0

Here, $1$1, strictly contradicting the conjectured lower bound.

The construction employs classical difference sets, specifically Singer sets in cyclic groups of order $1$2 for prime powers $1$3. The $1$4 points are indexed by such a difference set. Their geometric realization is in $1$5, where $1$6, via a weighted product of regular $1$7-gons. Pairwise distances correspond to a parameterized family determined by the cyclic separation of indices with a carefully designed profile for the weights—specifically, the sum of one low-frequency positive coefficient and higher-order decaying terms to ensure concavity and monotonicity in the distance sequence.

Scaling is performed so that the minimal gap between distances is $1$8, and the resulting largest distance (diameter) thus determines the key upper bound.

Analysis of the Separated Distance Profile

The profile for the set of pairwise distances is derived as follows. For each $1$9, the squared distance between two points whose cyclic separation is $1$0 is

$1$1

where

$1$2

with $1$3 exploited as a critical value to ensure the needed strict concavity and monotonicity for the resulting sequence $1$4.

Critical to the construction is the identification, via concavity, of the location and size of the minimal distance gap $1$5. Scaling the configuration so this difference equals $1$6 ensures all required separation constraints, with the overall diameter then being $1$7. The main technical content involves a precise asymptotic analysis, showing that as $1$8,

$1$9

contradicting the anticipated quadratic lower bound with constant $d=1$0.

Optimality and Extensions

The paper's explicit construction, based on a single-frequency perturbation, yields the constant $d=1$1. By adjusting the leading coefficient or using multiple frequencies, this constant can be further reduced. For example, optimizing the first coefficient up to

$d=1$2

gives a value near $d=1$3 for the leading factor. While numerically optimizing multiple frequencies can yield further improvements (e.g., $d=1$4), this has purely computational relevance and is not the focus of the current work.

The result does not address fixed, low dimensions. The open problem remains whether for each fixed $d=1$5, an $d=1$6 lower bound for the diameter persists for $d=1$7-point sets in $d=1$8 with $d=1$9-separated pairwise distances.

Implications and Future Directions

The main implication is the demonstration that in sufficiently high dimensions, the constraint of separation in the pairwise distance set does not enforce the conjectured quadratic diameter lower bound with constant $n$0. This reveals a surprising degree of geometric flexibility in high-dimensional Euclidean spaces, even under strong distance separation constraints. The explicit, formalizable constructions also provide new families of extremal configurations for point sets under separation constraints, potentially impacting further developments in finite geometry, additive combinatorics, and metric embedding theory.

From a methodological perspective, the use of formal proof assistants (Lean 4) to verify the correctness of the construction and arguments sets a rigorous standard for future work in extremal combinatorics and theoretical geometry.

It remains a central open challenge to determine the correct asymptotic in fixed low-dimensional settings, as well as to optimize further the separation profile to potentially improve diameter bounds and to classify all possible extremal configurations.

Conclusion

This paper provides a constructive refutation of Erdős’s conjecture on the diameter of sets with separated pairwise distances in high-dimensional Euclidean spaces, giving explicit configurations with diameter bounded above by $n$1. The work opens up new questions for low-dimensional cases and suggests that separation constraints are significantly less restrictive in high dimensions than previously conjectured. The rigorous, formalized approach further underlines the value of proof assistants in combinatorial geometry.