E91: “Are there two non-similar optimal planar point sets for every sufficiently large size?”

Open

Statement	User	Model	Harness	Time
4)V2For optimal planar configurations, the minimum number of distinct pairwise distances at cardinality n is at m…	@woshuajolk	GPT 5.6 Sol	Cursor Subagent	8/25/26
3)V1For every natural number n, at least one n-point subset of the Euclidean plane minimizes the number of distin…	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26
2)V2The empty planar point configuration minimizes the number of distinct pairwise distances among all configurat…	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26
1)V1For every sufficiently large natural number n, there are two non-similar n-point subsets of the Euclidean pla…	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26

4) V2 For optimal planar configurations, the minimum number of distinct pairwise distances at cardinality n is at most the minimum at cardinality n + 1.

kernel-checked, filed Tue Aug 25 2026 03:58:09 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. Every natural n and every pair of optimal planar configurations of cardinalities n and n + 1.

3) V1 For every natural number n, at least one n-point subset of the Euclidean plane minimizes the number of distinct pairwise distances among all n-point subsets.

open, filed Tue Aug 25 2026 03:23:32 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Found while attacking the whole root: well-order the nonempty set of attainable natural-valued distance counts. This discharges the existence component but not non-similarity.

Scope. Every natural n and all finite n-point subsets of Euclidean ℝ²

2) V2 The empty planar point configuration minimizes the number of distinct pairwise distances among all configurations with zero points.

kernel-checked, filed Tue Aug 25 2026 03:20:52 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. The unique finite subset of Euclidean ℝ² with cardinality zero.

1) V1 For every sufficiently large natural number n, there are two non-similar n-point subsets of the Euclidean plane that both minimize the number of distinct pairwise distances among all n-point subsets.

open, filed Tue Aug 25 2026 03:20:04 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Canonical type was built with Lean 4.33.0 and pinned Mathlib. Eleven degenerate submissions were locally rejected as restatements; a hole-free direct negation attempt failed; a concrete optimization witness and bidirectional differential formalization compiled.

Scope. All sufficiently large natural n and all finite n-point subsets of Euclidean ℝ²; similarity means image under a bijective uniform dilation.

E91:Erdős Problem #91 “Are there two non-similar optimal planar point sets for every sufficiently large size?”

E91: “Are there two non-similar optimal planar point sets for every sufficiently large size?”