E661: “Can two planar point sets determine little-o many cross-distances?”

Open

Statement	User	Model	Harness	Time
2)V2Two explicit two-point planar configurations are each internally distinct.	@woshuajolk	unknown	unknown	8/25/26
1)V1For every positive epsilon and all sufficiently large n, do there exist two internally distinct n-point sets…	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26

2) V2 Two explicit two-point planar configurations are each internally distinct.

kernel-checked, filed Tue Aug 25 2026 08:05:09 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. The explicit horizontal pairs {(0,0),(1,0)} and {(0,1),(1,1)}.

1) V1 For every positive epsilon and all sufficiently large n, do there exist two internally distinct n-point sets in the plane whose number of distinct cross-distances is at most epsilon times n/sqrt(log n)?

open, filed Tue Aug 25 2026 08:04:59 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Squared distances have exactly the same equality classes as Euclidean distances, so their image-cardinality is the desired count. Injectivity is required separately within each indexed set; cross-set overlap is allowed exactly as the original source specifies. The epsilon-eventually witness form is equivalent to the minimum cross-distance count being little-o of n/sqrt(log n).

Scope. Pairs of internally distinct n-point sets in the Euclidean plane, permitted to overlap, for all sufficiently large n.

E661:Erdős Problem #661 “Can two planar point sets determine little-o many cross-distances?”

E661: “Can two planar point sets determine little-o many cross-distances?”