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Statement

Notation (pp. 134--135, Section 5). $x_1,\ldots,x_n$ are $n$ distinct points in the plane, $d_1>d_2>\cdots>d_m$ the distinct distances they determine, and $u_i$ the number of pairs $(x_\mu,x_\nu)$ at distance $d_i$; the paper's parenthesis prints the defining equation as "$d(x_\mu,x_\nu)=u_i$" [sic], for $=d_i$. The pairs are unordered: the paper notes $\sum_{i=1}^m u_i=\binom n2$ (p. 135).

Conjecture (printed p. 135, unnumbered). Quoted, because the problem page's statement rests on its wording: "I conjectured that for $n>4$ the set $u_1,\ldots,u_m$ cannot be a permutation of $1,2,\ldots,n-1$ unless the $x_i$'s are equidistant points on a line or a circle."

The paper prints only this direction; it does not define "equidistant points on a circle". What it records with the conjecture, all on p. 135:

• For $n=4$ the paper calls such a multiplicity profile "clearly possible", by the three vertices of an isosceles triangle and the centre of its circumscribed circle.
• The conjecture fails for $n=5$, by an example of Pomerance: the vertices $x_1,x_2,x_3$ of an equilateral triangle, its centre $x_4$, and for $x_5$ one of the points where the circumscribed circle of $x_1,x_2,x_3$ meets the perpendicular bisector of the segment $(x_3,x_4)$.
• It also fails for $n=6$, by a communication from L. Berkes, a high-school student in Kecskemét; no configuration is printed.
• Erdős states that he is nevertheless fairly sure the conjecture holds for sufficiently large $n$, perhaps for all $n>5$.

Two questions follow it on p. 135. First, how many distinct values the $u_i$ can take: at most $n-1$, the paper notes, and perhaps for large $n$ that many only when the points are equidistant on a line. Second, the paper restates the conjecture as: if the $u_i$ are all distinct, then for $n\ge7$, $m$ cannot be $n-1$ unless the points lie on a line or circle; and it asks, assuming the conjecture, for the largest possible $m$ when the $u_i$ are all distinct. (In the corpus's words, the two forms agree: $n-1$ distinct positive integers summing to $\binom n2$ must be $1,\ldots,n-1$.)

Source. P. Erdős, Some old and new problems in combinatorial geometry, Annals of Discrete Math. 20 (1984), North-Holland Math. Stud. 87, pp. 129--136; the notation at the foot of p. 134 and the conjecture, its examples and the two questions on p. 135. The copy read is identified on the source card.

Read depth. Claims checked: the notation, the conjecture, the $n=4$ example, the two counterexample reports and the two questions were read clause by clause on the page images of pp. 134--135. The paper gives no verification of Pomerance's example and no details of Berkes's; neither was checked here. Nothing here is independently reviewed.

Proof pointer

A conjecture; the paper proves nothing about it and reports that it fails for $n=5$ and $n=6$.

Dependencies

None.

Bears on

• Problem 958: the conjecture is the problem's source, posed here for $n>4$ and counting unordered pairs; the paper itself reports failures at $n=5$ (Pomerance) and $n=6$ (Berkes) and expects it to hold for large $n$. The page's Statement asks the characterization for every $n$; the paper says nothing about its standing beyond these reports.