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Source. K. Vesztergombi, On large distances in planar sets, Discrete Math. 67 (1987), no. 2, 191--198, doi:10.1016/0012-365X(87)90027-6; the construction on pp. 197--198 with Fig. 7 (p. 197), read on the page images of the print. The edition is identified on the source card.

Read depth. Claims checked: the construction was read clause by clause on the page images. The paper asserts the count and gives no verification; none was carried out here. Nothing here is independently reviewed.

Statement

The notation is that of the Theorem on p. 192: $d_2$ is the second-largest distance of a planar set and $n_2$ the number of pairs at distance $d_2$.

Construction (pp. 197--198, unnumbered). Let $n=2m$. The outer points $v_1,\ldots,v_m$ are the vertices of a regular $m$-gon, among which, the paper states, $d_2$ occurs $m$ times. Points $u_1,\ldots,u_m$ are placed inside the circumscribed circle of the $v_i$ so that

$$d(v_i,u_i)=d(u_i,v_{i+1})=d_2,$$

indices taken modulo $m$. The paper concludes (p. 198) that these $2m$ points have $n_2=3m$, equality in the bound $n_2\le\frac32n$.

The paper states no range for $m$, and Fig. 7 (p. 197) draws the case $m=5$. The clause that $d_2$ occurs $m$ times among the $v_i$ needs a regular $m$-gon with at least two distinct distances, so $m\ge4$. The paper does not check that the added points leave $d_1$ and $d_2$ the two largest distances of the whole set.

Dependencies

None within the paper beyond the definitions of $d_2$ and $n_2$ (p. 191).

Bears on

• Problem 132: as asserted, the construction gives sets of $n=2m$ points whose second-largest distance occurs between $\frac32n$ pairs, more than $n$, so the two largest distances do not in general give the two distances the problem's first question asks for. It is not a counterexample to that question: the paper does not count the pairs at the smaller distances of these sets.