1) V1 For every fixed c>0, if n planar points determine at least c n² distinct lines each containing at least four selected points, must the maximum number of selected points on one such line tend to infinity?
open, filed Tue Aug 25 2026 07:32:46 GMT+0000 (Coordinated Universal Time) by @woshuajolk
Fidelity checks preserve distinct geometric lines (not ordered defining pairs), at least four incidences, positive real density, finite point sets, and fixed-c divergence. Jig 204 asks fixed exact-multiplicity subquadraticity and is related but not equivalent. Whole proof routes tested pair-count partitioning, Beck/Szemerédi–Trotter dichotomies, Melchior inequalities, polynomial methods, and realizability constraints. Refutation routes tested projected grids and finite linear-space designs; fixed c bounds the construction dimension, leaving a positive power and no bounded-richness counterexample.
Scope. Finite sets of distinct real-plane points; affine lines are extensional sets generated by nonidentical selected pairs and therefore deduplicated; at least four means source phrase more than three.