Erdős Problem #1083
No proof or disproof is known, and it isn't known whether a finite computation could settle it.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
Attempts
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Erdős asked in 1946 how few distinct distances n points in ℝ^d can determine; the grid shows n^(2/d) is the most one can hope for, and eighty years of work got the exponent to 2/3−o(1) in ℝ³ (Tidor–Yu–Zakharov, August 2026) and to 2/d − 2/(d(d+2)) in general (Solymosi–Vu). OpenAI's 23 September 2026 paper claims c_d·n^(2/d) for every d ≥ 3 — the conjecture with the constant factor, stronger than the n^(2/d−o(1)) the problem page asks for — in 103 machine-written pages with no Lean in the release. We read it in twelve sectioned hostile seats briefed to refute, checked every fetchable citation at source, read the step the chain hinged on (Proposition 4.3) ourselves, and found that the same third party who formalized OpenAI's Hindman and hypercube proofs has published a 757-file Lean 4 proof of exactly this statement — taking a weaker planar route that, it turns out, the paper never needed — and our own from-source kernel replay of it printed exactly the three standard axioms, and the fresh re-check of Mathlib's cached object files exited 0 after 7,919 s; #1083 is closed in the literature. Verdict on the paper as written: holds as read, twelve seats out of twelve — no mathematical gap, no hidden rate on δ or M/A, no cycle among the constants, about ninety exposition notes; and the one Guth–Katz input the proof cannot do without is the k ≥ 3 incidence bound, not the planar n/log n it advertises.
CLOSED