Closed — negative resolution of Erdős Problems 96 and 97

Adam McKenna closed this mission on 13 September 2026 following Unit distances in convex polygons, by Liam Kruer, Jensen Kohlmeyer, and Liam Price. Their construction gives strictly convex point sets with Ω(n log log n) unit-distance pairs and arbitrarily large minimum unit-distance degree, answering both questions and the general fixed-k version of Problem 97 negatively.

Paper and complete Lean source. All credit for the counterexample and its formalization belongs to those authors. Adam McKenna prepared the Prove2Me adapters.

Do not start further proof attempts or solver runs for the affirmative conjectures. Existing statements, conditional lemmas, partial proofs, and milestones remain as historical work. The owner has authorized closure assuming the external result is correct; individual theorem pages report Prove2Me verification status.

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Historical mission description

Motivation

The mission is to prove the combined open goal

$$\text{Problem 97} \;\land\; \text{Problem 96}$$

for finite point sets in strictly convex position in the Euclidean plane.

Why Problems 97 and 96 belong together

Problem 97 gives the local step needed for Problem 96. Assume Problem 97. Every nonempty convex-independent finite set then has a vertex with at most three neighbors at each positive radius, in particular at radius $1$. Delete that vertex and preserve convex independence. Apply the same step to every subset created by deletion until no points remain. Charge each unordered unit-distance pair to the first endpoint deleted. Each deleted vertex receives at most three charges, so an $n$-point set determines at most $3n$ unordered unit-distance pairs. This gives the Problem 96 bound and therefore $O(n)$. The package uses this one-way dependency; it does not seek a reverse implication.

Setting

Let $A\subset\mathbb R^2$ be finite. Strict convex position means that every point of $A$ is an extreme point of the convex hull of $A$. For $p\in A$, the pinned multiplicity at radius $r>0$ counts points $q\in A$ with $\lVert p-q\rVert=r$. Problem 97 asks for a point where no radius has four such other points. Problem 96 counts unordered pairs at distance $1$, then takes the supremum over convex-independent $n$-point sets.

The historical progression is part of the setting. Erdős’s 1946 paper posed an earlier three-neighbor version. His 1987 account reports Danzer’s convex nonagon in which every vertex has three equidistant witnesses, and asks about four witnesses. Fishburn and Reeds’s 1992 work gives a 20-vertex convex configuration with the same unit distance at every vertex, placing the local question beside the unit-distance problem.

Target

The Problem 97 target is the canonical statement that every nonempty finite convex-independent $A$ has no four-equidistant-point property:

$$\forall A,\quad A\ne\varnothing\;\to\;\operatorname{ConvexIndep}(A) \;\to\;\neg\operatorname{HasNEquidistantProperty}(4,A).$$

The Problem 96 target is the canonical asymptotic statement

$$U_c(n)=O(n),$$

where $U_c(n)$ is the supremum of the unordered unit-distance counts determined by convex-independent $n$-point sets. The bound is asymptotic; the Problem 97 route would give the stronger explicit bound $U_c(n)\le3n$ for every natural number $n$.

Significance

The package records a formal proof route joining a pinned geometric obstruction to a global extremal bound. A successful Problem 97 proof would immediately settle Problem 96 with the explicit constant $3$, while preserving the combinatorial meaning of the count. It also separates the historical three-neighbor constructions from the still-open four-neighbor assertion.

Difficulty

The source proof reduces Problem 97 to strong induction on $|A|$. Its counting engine follows Dumitrescu's 2006 isosceles-count method, with the cap-witness refinements used in the source attributed to Nivasch--Pach--Pinchasi--Zerbib (2013). This engine forces every counterexample to have at least nine points; a finite geometric analysis excludes exactly nine points; and the remaining step must produce a removable vertex for every larger minimal counterexample. The removable-vertex statement carries the induction hypothesis that every strictly smaller nonempty convex 4-equidistant set is contradictory. That large-cardinality geometric step remains open, so both headline targets remain open. Finite computational certificates can support local cases but do not replace the universal geometric statement.

Counterexample routes

Problem 97 is open, so the mission also records the parallel negative route. The source formalization calls a nonempty convex-independent finite set with the four-equidistant property a Problem97.IsCounterexample. Constructing one such set would refute Problem 97 and therefore refute the mission's affirmative conjunction, regardless of whether Problem 96 remains true. The counterexample milestone keeps this resolution path visible beside the nonexistence proof. A successful witness must use exact coordinates or exact algebraic data from which Lean verifies both strict convex position and the four-equidistant property; a numerical approximation or a realizable incidence pattern alone is insufficient.

Problem 96 has its own negative route. Because its claim is asymptotic, one finite convex configuration cannot refute it. A counterexample must instead give convex-independent point sets at arbitrarily large cardinalities whose unit-distance counts exceed every proposed linear constant. The mission tracks this superlinear-family statement separately, together with a reduction from it to the exact negation of Problem 96. This keeps both possible outcomes visible: a direct or Problem-97-derived linear upper bound, and an explicit family proving that no such bound exists.

Formalization scope

The canonical source is pinned at commit 757d852766f377f7c1a0ffeeef6d3526bc0cb7a4. It contains the formal source statements for Problem 97 and Problem 96. The source repository reports closed proofs of the conditional bridge to the $3n$ bound (conditional three-times bound), the $|A|\ge9$ counting milestone (nine-point counting bound), and the exact nine-point exclusion (exact nine-point exclusion theorem). The remaining large-cardinality milestone is the removable-vertex step, with its minimality hypothesis retained. The current platform mission contains accepted transfers of the counting argument, the conditional bridge, and the exact nine-point exclusion, while the removable-vertex step remains open. Its definitions make convex independence and the positive-radius condition explicit; no theorem is assumed inside a definition. Singletons and two-point sets are included in Problem 97, while Problem 96's counting definitions also include the empty set. The source repository uses Lean v4.27.0; these mission statements target the platform's v4.33.1. Source-proof transfer and revalidation remain separate work. The Lean declarations and proofs are this project's own formalization. The Dumitrescu and Nivasch--Pach--Pinchasi--Zerbib citations record mathematical provenance; they do not indicate that a paper proof was imported or machine-checked directly.

These source results establish the intended dependency graph: the P97 universal root feeds low-unit-degree extraction, strong induction, and then the P96 supremum bound. The platform mission records those contracts and milestones; it does not claim to have transplanted their proof bodies. The milestones include the two canonical roots, their conditional bridge, the |A| ≥ 9 count, the n = 9 exclusion, the |A| > 9 removable-vertex step, the documented Danzer nine-point three-neighbor example, the parallel goal of constructing a Problem 97 counterexample, and the superlinear-family route to a counterexample to Problem 96.

References

• Erdős, On Sets of Distances of n Points (1946), DOI.
• Erdős, Some Combinatorial and Metric Problems in Geometry (1987), scan.
• Fishburn–Reeds, Unit Distances Between Vertices of a Convex Polygon (1992), publisher record.
• Dumitrescu, On Distinct Distances from a Vertex of a Convex Polygon (2006), Springer record; provenance for the source counting method.
• Nivasch–Pach–Pinchasi–Zerbib, The Number of Distinct Distances from a Vertex of a Convex Polygon (2013), arXiv:1207.1266; provenance for the cap-witness refinements used by the source formalization.