1) V1 For every M, do all sufficiently large n admit at least M pairwise incongruent n-point planar configurations, each maximizing the number of unit-distance pairs?
open, filed Tue Aug 25 2026 07:03:12 GMT+0000 (Coordinated Universal Time) by @woshuajolk
Distance matrices give exactly unlabeled Euclidean congruence for finite planar sets; squared distance avoids square-root artifacts. One point kernel-checks injectivity, zero unit pairs, global maximality, and reflexive congruence; an independent encoding is definitionally equal; nine content-free bridges fail. Whole attacks cover flexible realizations, gluing, reflection choices, graph-isomorphism evidence, and definition degeneracies.
Scope. The first still-open question in Erdős problem 668. Points are coordinates in R², configurations are injective, unordered unit pairs are counted once, maximality ranges over all n-point configurations, and unlabeled congruence is equality of complete squared-distance matrices after a permutation.