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Statement

Setting (pp. 1--2). For a finite point set $P$, $u(P)$ is the number of unordered pairs of points of $P$ at Euclidean distance $1$. The paper calls a planar set $P$ in general position when $P$ contains no three collinear points and not the vertex set of a parallelogram, and sets $u'(n)=\max\{u(P):P\subset\mathbb R^2,\ \#P=n,\ P\text{ in general position}\}$.

Theorem 2 (p. 2, quoted). "There exists $c>0$ such that for any $n\ge2$, $u'(n)>cn\sqrt{\log n}$."

The paper notes that the grid and the Minkowski sum constructions for the unrestricted unit distance problem are unavailable under this condition, and that Brass observed a planar analogue of the construction of Erdős, Hickerson and Pach giving $u'(n)>cn\log^*n$ (p. 2). It knows no upper bound better than $u'(n)\le u(n)<cn^{4/3}$ (p. 2). The proof gives the asymptotic form $u'(n)>(1+o(1))\tfrac{\sqrt2}{4}n\sqrt{\log_2 n}$ (p. 4), the constant being smaller than for Theorem 1 because unordered pairs replace ordered pairs.

Proof pointer

Section 3 (pp. 4--6). The construction of Theorem 1 is repeated in the plane with rotations about the point $(0,r)$, $r>0$ large, in place of rotations of the sphere, and horizontal translations as their limit $r\to\infty$; the index sets are now subsets of the unordered pairs of $A$, where $A$ is a set of $t\ge1$ points with distinct $y$-coordinates in a $1/10$-neighbourhood of $(0,-1)$.

• Claim 3 (p. 4): for every $t\ge1$ and every sufficiently large $r>0$, $A$ can be chosen so that the rotated copies are pairwise disjoint and the translation construction $B$ contains no three collinear points and no parallelogram vertex set (so printed; the proof ends by showing that the rotation construction $B_r$ is the required set).
• Disjointness follows the proof of Claim 1 (p. 4). For the general position condition (pp. 5--6), Observation 5 (p. 5) shows that the translation distances $\beta(p_i,p_j)$, $i<j$, are linearly independent functions of the coordinates; with a comparison of chord lengths about $(0,r)$ this rules out parallelograms in $B_r$ for large $r$; collinear triples are ruled out by differentiating a $3\times3$ determinant in the coordinates, which the paper presents as a sketch.

Read depth

Claims checked: the definitions, Theorem 2 and Claim 3 were read clause by clause on the page images of the author version named on the source card, and the proof in Section 3 was followed. Nothing here is independently reviewed.

Dependencies

Theorem 1: the proof repeats its construction and the proof of its Claim 1.

Source. K. J. Swanepoel and P. Valtr, The unit distance problem on spheres, in Towards a Theory of Geometric Graphs, Contemp. Math. 342, Amer. Math. Soc., Providence, RI, 2004, 273--279, doi:10.1090/conm/342/06148; page numbers refer to the author version named on the source card.

Bears on

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