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18. Distinct distances under local condition

    1. Distinct distances under local condition

          Let $f(n,p,q)$ denote the smallest number of distinct distances generated by a set $P$ of $n$ points, under the condition that any $p$ points in $P$ determine at least $q$ distinct distances.

      Add remark

      Problem 18.1.

      [David Conlon] Is it true that $f(n,5,9) \leq o(n^2)$?
          Note: Recently, Tao proved that $f(n,4,5) \leq O\left( n^2 / \sqrt{\log(n)}\right)$.

      Reference: T. Tao. Planar point sets with forbidden 4-point patterns and few distinct distances. Preprint. arXiv:2409.01343.

          Cite this as: AimPL: Albertson conjecture and related problems, available at http://aimpl.org/albertson.