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Source. M. Charalambides, A note on distinct distance subsets, J. Geom. 104 (2013), no. 3, 439--442, DOI 10.1007/s00022-013-0176-0; read in the arXiv preprint arXiv:1211.1776v1, whose labels and page numbers are used here. Proposition 2.1 on p. 1; its proof on pp. 1--2; Remark 2.2 on p. 2. The journal version was not compared. The edition read is identified on the source card.

Statement

Definitions (p. 1): for a finite set $P\subset\mathbb R^2$, $\Delta(P)$ is the largest size of a subset $Q\subset P$ whose $\binom{|Q|}{2}$ pairwise distances are all distinct, and for a positive integer $N$, $\delta(N)$ is the minimum of $\Delta(P)$ over all $N$-element sets $P\subset\mathbb R^2$. The paper uses $\gtrsim$ without defining it; it is read here as $X\gtrsim Y$ meaning $X\ge cY$ for an absolute constant $c>0$.

Proposition 2.1 (p. 1). "$\delta(N)\gtrsim N^{1/3}/\log N$"

Remark 2.2 (p. 2). The paper states that, by scaling the probability in the proof, for every fixed $K>0$ there is a positive integer $N_K$ with

$$\delta(N)\ge\bigl(K-o_K(1)\bigr)\frac{N^{1/3}}{\log N} \qquad\text{whenever }N>N_K.$$

Since $K$ is arbitrary, the remark as printed says that $\delta(N)\log N/N^{1/3}\to\infty$. Conlon, Fox, Gasarch, Harris, Ulrich and Zbarsky, in Distinct volume subsets, credit the paper with $h_2(n)=\Omega(n^{1/3}/\log^{1/3}n)$, noting that the bound stated in the paper is slightly worse and that a careful analysis of its proof gives theirs.

Proof pointer

The proof (pp. 1--2) is Lefmann and Thiele's random selection with deletion. For $N>2$, keep each point of $P$ independently with probability $q$, then delete one point from each surviving isosceles triangle (ordered triples $(p,q_1,q_2)$ of distinct points with $\lVert p-q_1\rVert=\lVert p-q_2\rVert$, counted by $t$) and from each surviving quadruple of distinct points $(p_1,p_2,q_1,q_2)$ with $\lVert p_1-p_2\rVert=\lVert q_1-q_2\rVert$ (counted by $f$). The inputs are $t(P)\lesssim N^{7/3}$, which Pach and Sharir derived from the Szemerédi--Trotter theorem, and $f(P)\lesssim N^3\log N$ from the Guth--Katz theorem; both are cited, not proved. The choice $q=N^{-2/3}(\log N)^{-1}$ makes the expected size of the remaining set at least $\frac{N^{1/3}}{\log N}\bigl(1-a(\log N)^{-3}-b(\log N)^{-3}\bigr)$ for absolute constants $a,b>0$.

Coverage

Claims checked: the definitions, Proposition 2.1 and Remark 2.2 were read clause by clause on the page images of pp. 1--2. The deletion argument was read line by line; the cited bounds of Pach and Sharir and of Guth and Katz were not checked, and Remark 2.2's scaling was not carried out. Nothing here is independently reviewed.

Bears on. #1208, for $d=2$: $\delta(N)$ is that problem's $F_2(N)$, so the proposition gives $F_2(N)\gtrsim N^{1/3}/\log N$. It does not determine the order of $F_2(N)$; the upper bound the paper records is Proposition 1.2.