The weak pinned Erdős distance conjecture: some point determines n^(1-o(1)) distinct distances (Erdős Problem 604, first part)

Erdős problem #604 · erdosproblems.com/604

For a finite set $P\subset\mathbb R^2$ and a pin $x\in P$ let $D_x(P)=\{\|y-x\|:y\in P\setminus\{x\}\}$ . The integer grid shows that every pin can see as few as order $n/\sqrt{\log n}$ distances. Erdős asked (1957, Problem 16, and repeatedly later; erdosproblems.com Problem 604, a 500 dollar problem) whether every set of $n$ points in the plane has a point $x$ with $|D_x(P)|\gg n^{1-o(1)}$ , or even $\gg n/\sqrt{\log n}$ . Guth and Katz settled the unpinned count up to $\log n$ , but for pinned distances the best exponent was $0.8641\ldots$ (Katz-Tardos 2004). The weak pinned conjecture is the first form: for every $\varepsilon>0$ , does every sufficiently large $n$ -point planar set have a point determining at least $n^{1-\varepsilon}$ distinct distances?

Result: Proved(see note)
Status: Candidate (review pending)
AI contribution: AI-discovered
Method: Argument
Field: Combinatorial geometry; distinct distances
Posed by: Paul Erdős
Year posed: 1957
Years open: 69y
Solved: 2026-09-23
Model: Unreleased internal OpenAI model
Vendor: OpenAI
Collaborators: —
Verification: Lean-checked, statement unaudited
Publication: Announced
Collection: OpenAI math release (October 2026), version adc7f12
Significance: 42 / 100
Disclosed cost: —
Wikipedia: No dedicated article

What was actually shown

Theorem 1.1: for every fixed $s>0$ , $F_n(s)\to0$ , where $F_n(s)$ is the largest fraction, over $n$ -point planar sets, of ordered distinct pairs $(x,y)$ such that at least $n^s$ points of the set lie at distance $\|y-x\|$ from $x$ . Corollary 1.2: for every fixed $\varepsilon>0$ the fraction of pins with fewer than $n^{1-\varepsilon}$ distinct distances tends to $0$ uniformly, so every large set has such a pin, and in fact all but $o(n)$ points are. No explicit rate is given. Not shown: the stronger form $\gg n/\sqrt{\log n}$ in Erdős's question, the averaged conjecture $\sum_x|D_x|\gg n^2/\sqrt{\log n}$ , or any quantitative form of the $n^{o(1)}$ loss.

What the AI did

The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. 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 manuscripts are authored 'OpenAI' and name no human author. The principal manuscript is 'The weak pinned planar distance theorem' (September 23, 2026); its family also contains a separate unit-distance manuscript of the same date, entered separately.

Verification

No independent mathematician has checked this yet. Checked here: Theorem 1.1 and Corollary 1.2 were read against Erdős's question as recorded on erdosproblems.com (Problem 604) and in the manuscript. Corollary 1.2 gives, for every fixed $\varepsilon>0$ , that all but $o(n)$ points of any $n$ -point planar set determine at least $n^{1-\varepsilon}$ distances, uniformly over configurations, with no rate. Lean: formalization.yaml lists comparator ComparatorChallenges/PinnedDistances.json, declaration OAI.WeakPinned.main in OAI/Geometry/PinnedDistances/Main.lean. The statement file was read here: main says that for each $s>0$ the supremum over $n$ -point planar sets of the fraction of ordered pairs $(x,y)$ whose distance fiber at $x$ has at least $n^s$ points tends to $0$ . That is the paper's Theorem 1.1, which is stronger than the headline; the headline follows by the five-line pigeonhole count printed as Corollary 1.2, which is not itself formalised. Not rebuilt here. The proof uses real-closed-field transfer to number fields and the product formula.

The weak pinned Erdős distance conjecture: some point determines n^(1-o(1)) distinct distances (Erdős Problem 604, first part)

What was actually shown

What the AI did

Verification

Sources

Discussion