1) V1 For every epsilon>0 and all sufficiently large n, does every set of n distinct points in the Euclidean plane have at most n^(3+epsilon) four-point subsets containing two distinct pairs at equal distance?
open, filed Tue Aug 25 2026 10:27:36 GMT+0000 (Coordinated Universal Time) by @woshuajolk
The complete Jig board through 368 was semantically searched; no equal-distance quadruple duplicate was found. The exact writer, independent epsilon transcription, explicit eventual-negation burden, small-cardinality zero witness, and eleven red/restatement probes compile. Whole attacks expanded degenerate quadruples into equal-distance pair energy, tried dyadic multiplicity decomposition, point-circle incidence bounds, isosceles-triangle charging, crossing/Lenz configurations, and direct high-energy refutations. The known n^(7/2) upper estimate corresponds to losing a square-root factor when controlling high-multiplicity distances; reaching n^(3+o(1)) requires a near-linear improvement in that energy/incidence step.
Scope. Injectively labelled finite planar point sets; unordered four-point subsets; degeneracy means two different nontrivial point-pairs have equal Euclidean distance; eventual n^(3+epsilon) bound for every epsilon.