There exist n-point planar sets where every subset A has a distance occurring ≥|A|²/n^{1−δ} times, confirming Erdős's 1980 isosceles-triangle conjecture and answering a repeated-distance question negatively.
Distinct Distances: Open Problems and Current Bounds
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abstract
We survey the variants of Erd\H{o}s' distinct distances problem and the current best bounds for each of those.
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2026 1verdicts
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The Minkowski grid has robustly many repeated distances
There exist n-point planar sets where every subset A has a distance occurring ≥|A|²/n^{1−δ} times, confirming Erdős's 1980 isosceles-triangle conjecture and answering a repeated-distance question negatively.