Claims

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1963_02_01_altman: Altman (Amer. Math. Monthly, 1963) proves that every convex n-gon determines at least the floor of n/2 distinct distances, so the first question holds for every set in convex position; refereed.

1997_10_01_erdos_fishburn: Eight points in the plane, no three on a line, from each of which only three distinct distances are seen, published by Erdős and Fishburn with credit to Harborth; the second question has a negative answer.

2026_02_25_deepmind: A Lean 4 proof, found by a DeepMind prover agent from the formal-conjectures statement, that eight points with no three on a line need not contain a point seeing four distinct distances; the second question's answer is no.

2026_08_31_sallerk: A dated note linked from the site's thread deduces from the published maximum planar k-distance sets that the first question holds for every n at most 15; unreviewed.

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