The higher-dimensional Erdos distinct-distances conjecture

For a finite set $P\subset\mathbb R^d$ let $\Delta(P)=\{|p-q|:p\ne q\in P\}$ . Erdos introduced the distinct-distances problem in 1946; the integer grid $\{1,\ldots,t\}^d$ shows that $n$ points can determine only $O(n^{2/d})$ distances when $d\ge3$ . Known lower bounds fell short of this exponent: $n^{0.5643}$ and $n^{2/d-2/(d(d+2))}$ (Solymosi-Vu), $n^{3/5}$ up to logarithms in $\mathbb R^3$ via Guth-Katz, and $n^{2/3-o(1)}$ in $\mathbb R^3$ (Tidor-Yu-Zakharov). For every fixed $d\ge3$ , is there $c_d>0$ such that every set of $n\ge2$ points in $\mathbb R^d$ determines at least $c_dn^{2/d}$ distinct distances?

Result: Proved(see note)
Status: Candidate (review pending)
AI contribution: AI-discovered
Method: Argument
Field: Combinatorial geometry; incidence geometry
Posed by: Paul Erdos (1946, 'On sets of distances of n points'); the higher-dimensional form as recorded by Bardwell-Evans-Sheffer and Tidor-Yu-Zakharov
Year posed: 1946
Years open: 80y
Solved: 2026-09-23
Model: Unreleased internal OpenAI model
Vendor: OpenAI
Collaborators: —
Verification: Unreviewed
Publication: Announced
Collection: OpenAI math release (October 2026), version adc7f12
Significance: 45 / 100
Disclosed cost: —
Wikipedia: No dedicated article

What was actually shown

Theorem 1.1: for every integer $d\ge3$ there is $c_d>0$ such that every set of $n\ge2$ distinct points in $\mathbb R^d$ determines at least $c_dn^{2/d}$ distinct distances, matching the integer grid up to the constant. No dependence of $c_d$ on $d$ is made explicit, nothing is claimed for pinned distances (many distances from one point), and the planar problem, whose conjectured order is $n/\sqrt{\log n}$ , is untouched beyond reusing the Guth-Katz bound. Priority: Aksoy Yazici earlier claimed the same constant-factor bound in all dimensions $d\ge3$ , but withdrew the manuscript because a step was wrong; it is not an opposite claim.

What the AI did

The release README says the results were produced by an unreleased internal OpenAI model with one fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region, whose write-up was human-edited). The manuscript is authored 'OpenAI' and names no human author. The family has a single manuscript; it proves its concentration theorem and the needed incidence bounds, including the planar Guth-Katz input, in appendices.

Verification

No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against the conjecture; it is the full constant-factor statement $|\Delta(P)|\ge c_dn^{2/d}$ for every $d\ge3$ with no position assumption, not an $n^{2/d-o(1)}$ bound. The proof runs a minimal-counterexample induction on dimension through rigid-motion flats, Hilbert function estimates and approximate complete intersections; it was not refereed here. The paper itself notes an earlier claim of the same bound (Aksoy Yazici) that was withdrawn as incorrect. No Lean formalization exists for this family.

The higher-dimensional Erdos distinct-distances conjecture

What was actually shown

What the AI did

Verification

Sources

Discussion