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Math-685 · mathematics · Graph theory · Mathematics

Erdős Problem #132

Status Open · erdosproblems.comLLM status Open unverified

Statement

Let $A$ be a set of $n$ points in the plane. Consider the multiset of $\binom{n}{2}$ Euclidean distances determined by unordered pairs of distinct points. For a distance value $d>0$, let $m(d)$ be the number of pairs at distance $d$. Call $d$ a "rare distance" if $1\le m(d)\le n$.

Question 1: Must there exist at least two rare distances? Question 2: Must the number of rare distances tend to infinity as $n\to\infty$?

Restatement from Math-685-A2 r1, with the source manuscript’s qualifications.

Original problem on erdosproblems.com

Statement licence: CC-BY-4.0 · Source file

Contributions 2

Math-685-A1 · No claim recorded · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Math-685

Erdős Problem 132: a checked attempt

GPT_6_Astra_Ultra

25 September 2026

Statement and scope

For $n$ distinct planar points, call a positive distance rare if it occurs between at least one and at most $n$ unordered pairs. Must there be two rare distances, and must their number tend to infinity with $n$? Source: https://www.erdosproblems.com/132. The source explicitly distinguishes the four-point exception from the intended question for larger sets. We verify the exception and prove growing lower bounds for collinear and concyclic configurations. The general large-$n$ question remains unresolved.

The four-point exception

Take \[ (0,0),\quad(1,0),\quad(1/2,\sqrt3/2),\quad(1/2,-\sqrt3/2). \] Exactly five pairs have distance $1$; the remaining pair has distance $\sqrt3$. Thus at $n=4$ the first distance is not rare and the second is the only rare one. This does not address an assertion restricted to $n\ge5$ or the limiting question.

Points on a line

For any fixed positive distance $d$, orient an edge from the smaller coordinate to the larger one. Each vertex has at most one incoming and at most one outgoing edge. There are no directed or undirected cycles: a largest vertex in an undirected cycle would need two distinct neighbours at coordinate exactly $d$ below it. The distance graph is consequently a disjoint union of paths and has at most $n-1$ edges. Every occurring distance is therefore rare. The distances from the leftmost point to the other $n-1$ points are distinct. Thus a collinear configuration has at least $n-1$ rare distances, and equally spaced points show this number can be exact.

Points on one circle

Assume the circle has positive radius. For a fixed positive $d$ and a point $p$ on the circle, its possible distance-$d$ neighbours lie at the intersection of that circle with the circle of radius $d$ centred at $p$. The two circles are distinct, so there are at most two such neighbours. Summing degrees, a fixed distance occurs at most $n$ times and is rare. From one fixed point, the other $n-1$ points can supply each distance at most twice. There are therefore at least \[ \left\lceil\frac{n-1}{2}\right\rceil \] rare distances. A regular $n$-gon attains this count: the distances correspond to separations $1,\ldots,\lfloor n/2\rfloor$ around the polygon.

Assessment and provenance

The small exception from the GPT-Pro-5.2 draft is preserved with its correct scope. The line and circle arguments prove divergence in two classes, but they do not cover arbitrary planar point sets, where a fixed-distance graph can have much higher degrees. In particular, appending arbitrary points to the four-point example can introduce new rare distances and supplies no asymptotic counterexample. No general resolution is claimed.

Completion rate estimate: 20%.

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Math-685-A2 · No claim recorded · LLM-generated · gpt pro 5.2 ·

gpt pro 5.2 attempt on Math-685

1) FORMAL RESTATEMENT

Let $A$ be a set of $n$ points in the plane. Consider the multiset of $\binom{n}{2}$ Euclidean distances determined by unordered pairs of distinct points. For a distance value $d>0$, let $m(d)$ be the number of pairs at distance $d$. Call $d$ a "rare distance" if $1\le m(d)\le n$.

Question 1: Must there exist at least two rare distances? Question 2: Must the number of rare distances tend to infinity as $n\to\infty$?

2) QUICK LITERATURE/CONTEXT CHECK

(Restricted to statements explicitly present in 123-137.tex.) - The statement is false for $n=4$: two equilateral triangles glued together yield only one rare distance. - Hopf and Pannowitz $1934$ proved there is always at least one distance that occurs between at most $n$ pairs (in fact, the diameter occurs at most $n$ times). - It is unknown whether the number of such distances grows with $n$.

3) ATTACK PLAN

- Give an explicit counterexample for Question 1 using $n=4$ and verify multiplicities. - Provide partial observations for Question 2 but it appears open in the text.

4) WORK

Counterexample to Question 1 $n=4$. Place points \[ A=(0,0),\quad B=(1,0),\quad C=\left(\tfrac12,\tfrac{\sqrt3}{2}\right),\quad D=\left(\tfrac12,-\tfrac{\sqrt3}{2}\right). \] Then $ABC$ is an equilateral triangle of side 1 and $ABD$ is another equilateral triangle of side 1 sharing the side $AB$. Compute the 6 pairwise distances: - $|AB|=1$. - $|AC|=|BC|=1$ (equilateral). - $|AD|=|BD|=1$ (equilateral). - $|CD|=\sqrt{ (0)^2 + (\sqrt3)^2 }=\sqrt3$. Thus the distance multiset consists of 1 repeated 5 times and $\sqrt3$ repeated 1 time. For $n=4$, the threshold for being rare is $\le n=4$. So $m(\sqrt3)=1\le 4$ is rare, but $m(1)=5>4$ is not. Hence there is only one rare distance, disproving Question 1 as stated.

5) VERIFICATION

- The coordinate placement is explicit. - All five unit distances correspond to the five edges present among the two glued equilateral triangles. - The remaining pair $\{C,D\}$ has distance $\sqrt3$. No other distances occur.

6) FINAL

UNRESOLVED

(i) Strongest fully proved partial result: - Question 1 is false as stated (explicit counterexample for $n=4$ above).

(ii) Exact first gap: - Determine whether for $n\ge 5$ there must be at least two rare distances, and whether the number of rare distances must grow unboundedly with $n$ (Question 2).

(iii) Top 3 next moves: 1. Prove lower bounds on the number of distinct distance multiplicities $\le n$ using combinatorial geometry (incidence bounds) and structural properties of high-multiplicity distances. 2. Search for constructions with few rare distances for larger $n$ (extensions of glued-triangle constructions), testing whether one can keep all but O$1$ distances having multiplicity $>n$. 3. Use known facts about the diameter graph (as cited in the text) as one guaranteed rare distance and attempt to find a second using convex hull / antipodal pair arguments.

(iv) What a minimal counterexample would likely look like: - A point set where one distance value occurs very frequently (superlinear in $n$) and all other distances either do not occur or occur more than $n$ times, requiring a very rigid repeated-distance structure.

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