Must some vertex of a convex polygon have no 4 other vertices equidistant from it? (Erdős #97)
Statement
Does every convex polygon have a vertex with no other $4$ vertices equidistant from it? Precisely: if $P$ is a convex polygon with vertex set $V$, must there exist a vertex $v\in V$ such that no four other vertices of $V$ are all at the same distance from $v$? A counterexample is a convex polygon in which EVERY vertex $v$ admits four other vertices equidistant from $v$ (the common distance is allowed to depend on $v$). Convexity is essential: the analogous statement for non-convex polygons is false.
Acceptance. FULLY RESOLVES: (a) a counterexample — an explicit convex polygon with exact (algebraic or rational) vertex coordinates, together with machine-checkable exact-arithmetic certificates that (i) the points are in convex position and (ii) for every vertex a named set of four other vertices lies at exactly equal distances from it; floating-point coordinates alone do not suffice; or (b) a proof that every convex polygon has a vertex with no four other vertices equidistant from it — machine-checkable (Lean, building on the existing formal statement) preferred, else a complete written proof. ADVANCES: (a) partial constructions with certificates, e.g., a convex polygon in which all but a stated small number of vertices have four equidistant vertices, strictly improving any construction stated in the background or cited literature; (b) a certified proof that any counterexample must have more than $N$ vertices, for explicit $N$, via exhaustive or structural arguments with reproducible code; (c) resolution of the common-distance variant for $k=4$ (every vertex having four others equidistant at one global distance) in either direction, precisely stated. Deliver the coordinates plus exact verification code, or the proof file, or the partial-construction data with certificates.
Background
A problem Erdős returned to for half a century [Er46b; Er61; Er75f, p.100; Er87b, p.175; Er90; Er92e; Er95; Er97e], with a $100 prize on offer; listed as open on erdosproblems.com/97 (fetched 2026-07-13, status 'falsifiable'). The history calibrates the difficulty: Erdős's original 1946 conjecture had $3$ in place of $4$, and that version is FALSE — Danzer constructed a convex polygon on 9 vertices in which every vertex has three others equidistant from it (with the distance depending on the vertex; the construction is explained in [Er87b]), and Fishburn and Reeds [FiRe92] constructed a convex polygon on 20 vertices in which every vertex has three others equidistant at one common global distance. In [Er75f] Erdős even claimed Danzer had disproved the statement for every constant $k$; since that claim never reappeared in his later papers, it is presumed to have been a mistake, and the $k=4$ question stands open. The motivation is Erdős #96 (erdosproblems.com/96), the problem of how often a single distance can occur among vertices of a convex $n$-gon: if the present problem holds with $k+1$ equidistant vertices forbidden, an induction gives an upper bound of $kn$ there. Without convexity the answer is no: for any $d$ one can embed the $d$-dimensional hypercube graph in the plane with all edges of unit length and thread a non-self-intersecting polygon through its vertices, giving every vertex $d$ equidistant neighbours (observation credited to Alexeev and Mixon on the source page). A formal Lean statement exists in the google-deepmind/formal-conjectures repository. The attacker's tool: structured numerical optimization hunting a convex configuration in which every vertex has four equidistant vertices — generalizing the Danzer and Fishburn–Reeds constructions — followed by exact algebraic certification of the coordinates; or combinatorial-geometric proofs that small convex polygons cannot be counterexamples.
References
| Ref | Source | Type |
|---|---|---|
| REF-01 | Erdős Problem #97 (T. F. Bloom) | website |
| REF-02 | Erdős Problem #96 — the motivating convex unit-distance problem (T. F. Bloom) | website |
| REF-03 | Formalized statement of Erdős #97 (google-deepmind/formal-conjectures, Lean 4) | website |
Investigations · 0
No published investigations yet. This problem is unclaimed territory.