Let $A$ be a set of $n$ points in $R^{2}$ such that all pairwise distances are at least $1$ and if two distinct distances differ then they differ by at least $1$ . Is the diameter of $A$ $≫ n$ ?

Open — best to date is a honest null, not yet sealed.

geometry · open · formalized (Lean) · 1 attempt

machinery: geometric,distinct-distances,Guth-Katz-polynomial-method,consecutive-integer-window,incidence-geometry,extremal-set-system

use this record

vela registry pull vfr_37aec80d874a0239

vela reproduce examples/erdos-problems

evidence

honest null

needs verification

attempted via frontier 'difference/covering' (transfer_strength=weak) -> no_progress

No solve/partial on this pass. Transfer into the owned frontier was 'weak'. Do not re-attack cold; needs a new idea or richer accumulated context.

unverified AI candidates (2)

gpt-erdos · GPT-5.2 Pro + Deep Research · unverified

This is **open** (it is an Erdős problem). ([Erdős Problems][1])

candidate solution ↗

llm-hunter · gpt pro 5.2 · unverified

1 LLM attack(s) recorded (gpt pro 5.2); unverified.

candidate solution ↗

formal

AMS 52 · open (literature)

theorem erdos_100 :
    answer(sorry) ↔ ∃ C > (0 : ℝ), ∀ᶠ n in atTop, ∀ A : Finset ℝ²,
      A.card = n →
      DistancesSeparated A →
      diam (A : Set ℝ²) > C * n

formal-conjectures/100.lean ↗

status

open

evidence

formal

status

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