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More Distinct Distances Under Local Conditions

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We establish the following result related to Erdős’s problem on distinct distances. Let V be an n-element planar point set such that any p members of V determine at least \(\left( {\begin{array}{*{20}{c}} p \\ 2 \end{array}} \right) - p + 6\) distinct distances. Then V determines at least \(n^{\tfrac{8} {7} - o(1)}\) distinct distances, as n tends to infinity.

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Correspondence to Andrew Suk.

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Supported by a Packard Fellowship, by NSF CAREER award DMS-1352121, and by an Alfred P. Sloan Fellowship.

Research partially supported by Swiss National Science Foundation grants 200020-165977 and 200021-162884.

Supported by NSF grant DMS-1500153.

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Fox, J., Pach, J. & Suk, A. More Distinct Distances Under Local Conditions. Combinatorica 38, 501–509 (2018).

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Mathematics Subject Classification (2000)