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Statement

Setting (Section 4, p. 170). For distinct points $x_1,\ldots,x_n$ in the plane, $D(x_1,\ldots,x_n)$ is the number of distinct distances they determine, and $g(n)=\min D(x_1,\ldots,x_n)$ over all such sets. Erdős recalls his conjecture (9), $c_1n/(\log n)^{1/2}<g(n)<c_2n/(\log n)^{1/2}$, whose upper bound comes from the lattice points and whose lower bound carries a prize offer. He then sets the value of $g(n)$ aside and asks for which $n$ a set attaining $g(n)$ is unique up to similarity.

Small cases (pp. 170-171), as Erdős reports them.

• $n=3$: unique, the equilateral triangle.
• $n=4$: not unique; $g(4)=2$, attained by a square and by two equilateral triangles sharing an edge.
• $n=5$: $g(5)=2$, and "it seems that the regular pentagon is the only solution"; Erdős adds that a detailed proof was given by a colleague from Zagreb, whose letter he does not have.
• $n=6,7,8$: $g(6)=g(7)=3$ and $g(8)=4$, and Erdős says it is easy to see that uniqueness fails.
• $n=9$: a colleague remarked that $g(9)=4$, attained by the regular nonagon. Gy. Hegyi found a second configuration: the six vertices of a regular hexagon, its centre, and the mirror images of the centre in two neighbouring sides.

Erdős had thought that for $n>5$ the value $g(n)$ can always be attained in more than one way; the nonagon raised the question for $n=9$, and Hegyi's example settled it.

Question (p. 171). "Is it true that for $n>n_0$, $g(n)$ can always be implemented in more than one way?" Erdős adds that he does not see how to attack it. In context, two ways are two attaining sets that are not similar.

Source. P. Erdős, Some combinatorial and metric problems in geometry, Intuitive geometry (Siófok, 1985), Colloq. Math. Soc. János Bolyai 48, North-Holland, Amsterdam-New York, 1987, 167--177 (MR 89i:52012); Section 4, display (9) on p. 170, the uniqueness discussion on pp. 170-171 and the question on p. 171.

Read depth. Claims checked: the definition of $g(n)$, display (9), the small cases and the question were read clause by clause on the page images of pp. 170-171. The small-case values are Erdős's reports; none is proved in the paper, and they were not re-derived here.

Proof pointer

None; the question is open in the paper, and the small cases are reported without proof.

Dependencies

None.

Bears on

• Problem 91: the question is the site's statement, that for sufficiently large $n$ there are at least two non-similar $n$-point sets minimizing the number of distinct distances; the site's "and probably many" is not in this paper's question.