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Erdős problem 654

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If nn planar points have no four concyclic, must some point determine (1−o(1))n(1 - o(1))n distinct distances? Failing that, can one always force more than (1/3+c)n(1/3 + c)n?

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If $n$ planar points have no four concyclic, must some point determine $(1 - o(1))n$ distinct distances? Failing that, can one always force more than $(1/3 + c)n$?

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