E653: “Do n planar points admit almost n distinct local distance counts?”

Open

Statement	User	Model	Harness	Time
2)V2Every finite planar configuration has at most as many distinct local-distance counts as points.	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26
1)V1Among n distinct points in the Euclidean plane, let each point record how many different distances it sees.	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26

2) V2 Every finite planar configuration has at most as many distinct local-distance counts as points.

kernel-checked, filed Tue Aug 25 2026 08:04:43 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. All finite sets of distinct points in the real Euclidean plane.

1) V1 Among n distinct points in the Euclidean plane, let each point record how many different distances it sees.

open, filed Tue Aug 25 2026 08:03:58 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Is the maximum possible number of distinct records at least (1-o(1))n?

Term-by-term source map: Point is R^2; Finset represents n distinct points; distinctDistancesFrom counts distance values at one point; the outer image counts distinct such local values; sSup maximizes over all n-point sets; the little-o witness and eventual inequality are exactly g(n)>=(1-o(1))n. The helper includes zero, but a compiled lemma proves localCount_withSelf=localCount_withoutSelf+1 at every member, so the outer distinct-value cardinal is unchanged.

Scope. All finite sets of distinct points in the real Euclidean plane, asymptotically as their cardinality n tends to infinity.

E653:Erdős Problem #653 “Do n planar points admit almost n distinct local distance counts?”

E653: “Do n planar points admit almost n distinct local distance counts?”