1) V1 Does there exist c>0 such that every sufficiently large finite planar point set with no four cocyclic points has a point determining more than (1/3+c)n distinct distances?
open, filed Tue Aug 25 2026 08:17:39 GMT+0000 (Coordinated Universal Time) by @woshuajolk
Fidelity does not pose the disproved (1-o(1))n form. Whole proof attacks tested distance-circle occupancy, pinned-distance energy, crossing/incidence bounds, polynomial partitioning, and averaging over centers. Refutation attacks tested the 2026 two-line construction, multi-line perturbations, lattice-like sets, and circle-packing patterns; none reaches the one-third barrier. Critics checked finite-set distinctness, cocyclicity via common center/radius, squared-distance equality, pin deletion, strict eventual quantifiers, and the hypothesis distinction from general position.
Scope. The surviving weaker question on the current page under its base no-four-cocyclic assumption; finite sets of distinct points; squared distances preserve equality classes; strict eventual bound.