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Statement

Setting (p. 1). $U(S)$ is the number of pairs of points of a planar set $S$ at Euclidean distance $1$, and $U_c(n)=\max U(\mathcal P)$ over all convex $n$-gons $\mathcal P$ in the plane, a polygon being identified with its vertex set.

Theorem 1 (p. 3, quoted). "For each positive integer $n$, $U_c(n)\le n\log_2 n+4n$."

The paper places this against Füredi's 1990 bound $U_c(n)\le 2\pi n\log_2 n+O(n)$, which it describes as the best upper bound known before it (p. 1), and against Erdős and Moser's conjecture $U_c(n)=\Theta(n)$ (p. 1).

Proof pointer

Section 2.2 (p. 5). Split $\mathcal P$ by an antipodal cut into two polygonal chains $\mathcal P_1$ (with $a$ vertices) and $\mathcal P_2$ (with $b$ vertices), $a+b=n$; the unit distances between the chains are the $1$-entries of the $a\times b$ cut matrix $\mathbf M_{\mathcal P}$, the skeleton of the matrix of distances between the two chains. Proposition 1 (p. 4, due to Brass and Pach, cited) gives $U(\mathcal P_1)+U(\mathcal P_2)\le 2n$. Proposition 3 (p. 4, proved there: every distance matrix has the obtuse angle property, see Theorem 2's page for the definition) shows that $\mathbf M_{\mathcal P}$ avoids a fixed $4\times4$ pattern $\mathbf C'$, which is the amalgam of a $3\times2$ pattern $\mathbf A$ and a $2\times3$ pattern $\mathbf B$. Keszegh's amalgam inequality (Lemma 1, p. 4, cited) and Tardos's bounds $\operatorname{ex}(a,b,\mathbf A)\le\frac{a+b}2\log_2(a+b)+2b$ and $\operatorname{ex}(a,b,\mathbf B)\le\frac{a+b}2\log_2(a+b)+2a$ (Lemma 2, p. 4, cited) give at most $n\log_2 n+2n$ unit distances across the cut, and adding the $2n$ of Proposition 1 gives the theorem.

Read depth

Claims checked: the definitions, Theorem 1 and Propositions 1--3 were read clause by clause on the page images of arXiv:1009.2216v3, and the proof in Section 2.2 was followed. Lemmas 1 and 2 and Propositions 1 and 2 are cited by the paper, not proved there, and their sources were not read. Nothing here is independently reviewed.

Dependencies

None in the corpus. External inputs named by the paper: the antipodal-cut bound of Brass and Pach (Proposition 1), Keszegh's amalgam lemma (Lemma 1) and Tardos's extremal bounds for a $3\times2$ and a $2\times3$ pattern (Lemma 2).

Source. A. Aggarwal, On unit distances in a convex polygon, Discrete Math. 338 (2015), no. 3, 88--92, doi:10.1016/j.disc.2014.10.009; the edition read is named on the source card.

Bears on

• Problem 96: the problem asks whether $U_c(n)=O(n)$; Theorem 1 gives the upper bound $U_c(n)\le n\log_2 n+4n$, which does not decide that question.