Pinned distinct distances with no four cocircular points
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For an n-point planar set with no four points on a circle, must some point determine nearly n distinct distances? The counterexample target gives a family where every point determines fewer than 3n/4.
Why it matters and what counts as progress
Why it matters: It tests whether the no-four-cocircular condition forces a strong pinned-distance conclusion.
What counts as progress: Explicit no-four-cocircular point sets violating the proposed pinned-distance bound, or a stronger lower bound valid under that geometric condition.
How the problem developed
- 2020 · Construction [1]
Adrian Dumitrescu Studied distinct distances under forbidden four-point patterns, including sets with no four cocircular points; this literature addressed global distance counts rather than the later pinned near-n question.
- January 2026 · Counterexample [2]
Feng and coauthors; Aletheia Reported n-point sets with no four cocircular points for which each point determines fewer than 3n/4 distances, refuting the near-n pinned-distance assertion under that condition.
AI contribution
Semi-Autonomous Mathematics Discovery with Gemini: A Case Study on the Erdős Problems
Aletheia constructed planar n-point sets with no four cocircular in which every point determines fewer than 3n/4 distinct distances. These refute the near-n-distances question under that hypothesis.
Sources
Website timeline
Publication dates and editorial changes.
Aletheia counterexample to an Erdős distinct-distance question