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Statement

Notation (pp. 134--135, Section 5), as on the conjecture's page: $x_1,\ldots,x_n$ are $n$ distinct points in the plane, $d_1>\cdots>d_m$ their distinct distances, and $u_i$ the number of unordered pairs at distance $d_i$, so that $\sum_iu_i=\binom n2$.

What the paper prints on p. 135, in the corpus's words:

• The regular $(2k+1)$-gon. Its vertices give $m=k$, with every $u_i$ equal to $2k+1$.
• Pannwitz's bound. By an old result of Pannwitz (cited by name only, with no reference), the diameter of $x_1,\ldots,x_n$ occurs at most $n$ times; hence $\min u_i\le n$, with equality when $n$ is odd and the points form a regular polygon.
• The question. Suppose all the $u_i$ are equal, to a common value $t_n$. The paper notes that $t_n=1$ is possible and that $t_n=n$ is possible if and only if $n$ is odd, both without proof, and that $t_n\le n$ by Pannwitz's result and $t_n\mid\binom n2$. It asks: "What values are possible for $t_n$?"

The paper prints no statement that among $n$ points some two distances, or some unbounded number of distances, each occur between at most $n$ pairs.

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, the rest on p. 135. The copy read is identified on the source card.

Read depth. Claims checked: the paragraph on p. 135 was read clause by clause on the page image. Pannwitz's bound is quoted by the paper without a reference or proof and was not checked against Pannwitz's work here. Nothing here is independently reviewed.

Proof pointer

A question. The bound $\min u_i\le n$ is the paper's one-line consequence of Pannwitz's diameter result, since the diameter $d_1$ is one of the $d_i$; the claims $t_n=1$ possible and $t_n=n$ possible exactly for odd $n$ are asserted without proof.

Dependencies

Pannwitz's result that the diameter of $n$ planar points occurs at most $n$ times, cited without a reference.

Bears on

• Problem 132: the site and Clemen, Dumitrescu and Liu attribute that problem to this paper ([Er84c]). The paper holds the facts above, among them the diameter bound that makes one distance of multiplicity at most $n$ always exist, and the regular $(2k+1)$-gon, in which every distance occurs exactly $n$ times; it prints no statement of the problem's question, so the attribution is not confirmed by this paper.