The higher-dimensional Erdős distinct-distances conjecture

Any n points in d ≥ 3 dimensions span at least a constant times n^(2/d) distinct distances, the paper claims.

1 paper September 23, 2026 Explained by AI

Place n points in space and list all the distances between pairs. Erdős asked how few different distances there can be. In d dimensions a grid gives about n^(2/d), and the conjecture says nothing does better by more than a constant factor. This paper claims to prove that for every fixed d of at least 3. No Lean formalization is listed.

First step

1946

Paul Erdős, who posed the distinct-distances problem

Significance

Major

If the proof holds, it settles the higher-dimensional form of a famous problem with the best possible power.

Trust

Claimed, not checked

Field

Combinatorics

1 paper

Best proven exponent for distinct distances among n points in four dimensions

Year	Who	Value
2008	Solymosi–Vu	0.4167
2026	Solymosi–Vu recurrence with Tidor–Yu–Zakharov	0.4706
2026	This paper	0.5

Higher is better. The conjectured exponent is 2/4 = 0.5. The 2026 bound of about 0.471 also loses an o(1) in the exponent.

The explanation

Build a cube-shaped grid of points, 10 on a side, so 1,000 points in all. Measure the distance between every pair. Squared distances are whole numbers from 1 up to 3 × 9², which is 243. A direct count finds only 144 different distances among almost half a million pairs. For comparison, 1,000 raised to the power 2/3 is 100.

That is the pattern behind the question. In d dimensions, a grid with n points spans about n^(2/d) distances, up to a constant factorSource 1. Erdős introduced the distinct-distances problem in 1946Source 2. The higher-dimensional conjecture says no set of n points can do better than a constant times n^(2/d)Source 3.

The paper claims exactly that boundSource 4. For each d of at least 3, there is a constant c_d above zero. Every set of n ≥ 2 points spans at least c_d × n^(2/d) distancesSource 4. No assumption is made on where the points sitSource 5. The constant depends only on the dimension, and the paper gives no number for it.

Earlier bounds fell short in the power. In three dimensions, Aronov, Pach, Sharir and Tardos reached about n^(77/141), roughly n^0.546Source 6. Solymosi and Vu built recurrences linking dimensionsSource 7. Their original theorem gives about n^0.5643 in three dimensionsSource 8. In four dimensions it gives n^(5/12), about n^0.417Source 8. In 2026, Tidor, Yu and Zakharov proved n^(2/3 − o(1)) in three dimensionsSource 9. Fed into the recurrences, that gives n^(8/17 − o(1)) in four dimensionsSource 10. That is about n^0.471, still short of 0.5Source 11.

An earlier manuscript by Aksoy Yazici claimed the same constant-factor bound. Its author withdrew it, because the proof of its Theorem 1.2 is not correctSource 12.

Why does the constant factor matter in three dimensions, where the power 2/3 was nearly reached? A bound like n^(2/3 − o(1)) still allows the ratio to the target to shrink forever, just very slowlySource 13. Ruling that out means excluding every slowly shrinking ratio, without first fixing a power savingSource 14.

The single hard part is concentration. Bounds from lower dimensions limit how many points can sit on one flat plane or lineSource 15. They do not stop points from crowding onto curved surfaces, and no single polynomial degree captures all such crowdingSource 15. The paper handles this by tracking crowding at many polynomial degree scales at onceSource 16. Its main step rules out a covering by thin cones [shapes swept out by lines through one point]Source 17.

The flat, two-dimensional case is different. There a square grid gives about n divided by the square root of log n distances, so the conjectured answer has another formSource 18. Guth and Katz proved a bound of n divided by log n in the plane in 2015. This paper uses that planar result as its starting inputSource 19.

How sure is this? No Lean formalization is listed for this paper in the openai/math release. So no computer proof checker has checked it. No specialist review has been published.

What it doesn't show. This does not touch the flat, two-dimensional case, which has a different conjectured answer, and it gives no numerical value for the constant.

Where it came from

This result comes from an unreleased OpenAI model, published in the openai/math release on 6 October 2026.

Hype check

Major only if the proof holds. The constant c_d is not given as a number, and an earlier claim of the same bound was withdrawn. No Lean formalization is listed and no specialist review is published.

Words used

Distinct distances: The number of different values among all distances between pairs of points in a set.
Constant factor: A fixed multiplier that does not grow or shrink with the number of points.
o(1) in the exponent: A small loss in the power that shrinks to zero as n grows, but may shrink very slowly.
Recurrence: An inequality that turns a bound in one dimension into a bound in a higher dimension.
Polynomial degree scale: The highest power allowed in the polynomial equations used to describe where points crowd.
Cone: A shape made of straight lines all passing through one point.

The original paper

The higher-dimensional Erdős distinct-distances conjecture (PDF, opens in a new tab)
September 23, 2026

All quoted lines (19)

The power of $n$ is optimal: the integer grid $\{1,\ldots,t\}^d$ has $t^d$ points, and its squared distances are integers between $1$ and $d(t-1)^2$ .
The higher-dimensional Erdős distinct-distances conjecture Back to text
Erd\Hos introduced the problem in 1946.
The higher-dimensional Erdős distinct-distances conjecture Back to text
The higher-dimensional conjecture predicts a constant multiple of $n^{2/d}$ distances among any $n$ points of $\mathbb R^d$ , for every fixed $d\ge3$ ; see, for example,.
The higher-dimensional Erdős distinct-distances conjecture Back to text
For every fixed integer $d\ge3$ , we prove that every set of $n\ge2$ distinct points in $\mathbb R^d$ determines at least $c_d n^{2/d}$ distinct distances, where $c_d>0$ depends only on $d$ .
The higher-dimensional Erdős distinct-distances conjecture Back to text
No assumption is made on the position, spacing, or concentration of the points.
The higher-dimensional Erdős distinct-distances conjecture Back to text
Aronov, Pach, Sharir, and Tardos improved the three-dimensional lower bound to $\Omega_\varepsilon(n^{77/141-\varepsilon})$ for each $\varepsilon>0$ and propagated improvements to higher dimensions.
The higher-dimensional Erdős distinct-distances conjecture Back to text
Solymosi and Vu developed two recursive inequalities connecting distance bounds in lower and higher dimensions.
The higher-dimensional Erdős distinct-distances conjecture Back to text
Their original theorem gives $\Omega(n^{0.5643})$ distances in dimension three and $\Omega_d(n^{2/d-2/(d(d+2))})$ for every fixed $d\ge4$ .
The higher-dimensional Erdős distinct-distances conjecture Back to text
More recently, Tidor, Yu, and Zakharov proved $n^{2/3-o(1)}$ distinct distances for arbitrary point sets in $\mathbb R^3$ .
The higher-dimensional Erdős distinct-distances conjecture Back to text
Inserting their three-dimensional distance bound into the Solymosi–Vu recurrences gives $n^{8/17-o(1)}$ in dimension four and $n^{3/8-o(1)}$ in dimension five.
The higher-dimensional Erdős distinct-distances conjecture Back to text
The leading exponents in these two deductions still fall below the conjectured value $2/d$ .
The higher-dimensional Erdős distinct-distances conjecture Back to text
An earlier manuscript of Aksoy Yazici claimed the constant-factor bound in all dimensions $d\ge3$ , but was withdrawn by the author because the proof of its Theorem 1.2 is not correct.
The higher-dimensional Erdős distinct-distances conjecture Back to text
The ratio $|\Delta(P)|/n^{2/d}$ could still tend to zero more slowly than every negative power of $n$ .
The higher-dimensional Erdős distinct-distances conjecture Back to text
To obtain a constant factor, the proof must exclude every such sequence, without first fixing a power saving.
The higher-dimensional Erdős distinct-distances conjecture Back to text
They do not prevent substantial concentration on more general varieties, nor do they identify a single polynomial degree at which all of that concentration can be studied.
The higher-dimensional Erdős distinct-distances conjecture Back to text
We use these two ideas to keep track of concentration at several polynomial degree scales.
The higher-dimensional Erdős distinct-distances conjecture Back to text
Theorem … rules out this coverage, including the choice of either endpoint for each pair.
The higher-dimensional Erdős distinct-distances conjecture Back to text
In the plane, the square-grid construction gives $O(n/\sqrt{\log n})$ distances; the planar problem consequently has a different conjectural order.
The higher-dimensional Erdős distinct-distances conjecture Back to text
That bound supplies the two-dimensional input to our induction.
The higher-dimensional Erdős distinct-distances conjecture Back to text

Other sources

167Planar distinct distances and unit-distance boundsCombinatorics
171The hypercube Ramsey conjectureCombinatorics
159Erdős’s reciprocal-sum conjecture and quasipolynomial Szemerédi boundsCombinatorics
181The Erdős–Gallai cycle-decomposition conjectureCombinatorics
084The geometric case of the Erdős similarity conjectureReal and complex analysis

Browse Combinatorics Erdős problems Named conjectures Distances and packing Open since the 1940s Claimed, not yet checked