Claims

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1995_06_01_erdos_fishburn: Every five-point and every six-point planar set has two distinct distances each occurring between at most n pairs: the cases n = 5 and n = 6 of the first question, by Erdős and Fishburn's small-case classifications; refereed.

2025_05_07_clemen_dumitrescu_liu: For n at least 5, every convex n-point planar set, and every n-point set with small enough first two convex layers, has a distance besides the diameter occurring at most n times (Theorems 1.2 and 1.3); refereed in Acta Math. Hungar.

2026_01_28_zeraoulia: Every seven-point planar set has two distinct distances each occurring between at most seven pairs, and an eight-point counterexample would have multiplicities (1, 9, 9, 9) with a unique diametral pair; a self-published note, claimed.

2026_07_05_marchetto: Every planar set of n points, 7 ≤ n ≤ 13, has two distinct distances each occurring at most n times, unconditionally except for n = 11 and 12, which rest on Wei's 11-point 5-distance list; a note with checking code, claimed.

2026_07_25_ienjoymath: An independent proof, checked by the claimant's script, of the first question for n = 7, 8, 9, 10 and 13, and a 9n/7 lower bound for the extremal problem of Clemen, Dumitrescu and Liu; an anonymous thread-posted note, claimed.

2026_08_23_beller: Every eight-point planar set has two distinct distances each occurring between at most eight pairs: the case n = 8 of the first question, deduced in Lean from three published results; a forum proof claim, claimed.

2026_09_25_jones: Under a convex-layer condition the diameter and second-largest distance each occur at most n times, and the smaller multiplicity of the second-largest and smallest distances is at most 4n/3 plus a constant; a forum claim, claimed.

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