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Statement

Setting (pp. 2--5, 7). $m_1(\mathbb R^2)$ is the supremum of the upper densities of measurable planar sets with no two points at distance $1$ (p. 2). $T_{l_1,l_2,\alpha}$ is the flat torus spanned by $\vec v_1,\vec v_2$ with $|\vec v_1|=l_1$, $|\vec v_2|=l_2$ and angle $\alpha\in(0,\pi/2]$, $\rho$ is its metric induced by the Euclidean metric (Definitions 3 and 4, p. 3), and the torus is perfectly periodic when $\rho(p_1,p_2)\ne1$ implies that no lattice translate of $p_2-p_1$ has length $1$ (Definition 5, p. 5). $\operatorname{diam}(F)$ is the diameter of $F\subset T_{l_1,l_2,\alpha}$ in $\rho$.

Let $T_{l_1,l_2,\alpha}=F_1\sqcup\cdots\sqcup F_n$ be a partition of a perfectly periodic torus into measurable sets with $\operatorname{diam}(F_i)<1$ for every $i$, and let $p_i\in F_i$ be arbitrary points. Let $G=(V,E)$ be an undirected graph on $V=\{p_1,\ldots,p_n\}$ whose edge set satisfies, for all $1\le i<j\le n$,

$$(p_i,p_j)\notin E\ \Longrightarrow\ \rho(q_1,q_2)\ne1\quad\text{for all }q_1\in F_i,\ q_2\in F_j$$

(p. 7).

Lemma 3 (p. 7). If $M\subset\{p_1,\ldots,p_n\}$ is an independent set of $G$, then

$$m_1(\mathbb R^2)\ge\frac{\sum_{p_i\in M}\lambda_2(F_i)}{\sum_{j=1}^n\lambda_2(F_j)},$$

where $\lambda_2$ is planar Lebesgue measure.

The edge condition is one-sided: any graph with at least the forced edges qualifies, and the paper notes that fewer edges lead to a larger maximum independent set (p. 8).

Source. Alexander Tolmachev, On lower bounds of the density of planar periodic sets without unit distances, arXiv:2411.13248v2 (11 Apr 2025): Lemma 3 and the graph condition on p. 7, its proof on pp. 7--8, Definitions 3--5 on pp. 3 and 5. The edition read is identified on the source card.

Read depth. Claims checked: the statement, the graph condition and the definitions it uses were read clause by clause on the printed pages, and the proof was read. Nothing here is independently reviewed.

Proof pointer

Pages 7--8. Let $A$ be the union of the pieces $F_i$ with $p_i\in M$ and $\hat A$ its periodic extension to the plane by the lattice spanned by $\vec v_1,\vec v_2$; $\hat A$ is measurable with density $\lambda_2(A)/\lambda_2(\text{torus})$. Two points of $\hat A$ at Euclidean distance $1$ project to points of one piece, at torus distance below $1$, or of two non-adjacent pieces, at torus distance not $1$; in both cases perfect periodicity rules out a Euclidean distance of $1$ between any lattice translates, so $\hat A$ avoids distance $1$.

Dependencies

Definitions 3--5 (pp. 3, 5) of the same paper. Used for Theorem 1 (p. 10), where the pieces are the $nm$ equal hexagons of the grid.

Bears on

• Problem 1070: the lemma gives lower bounds on $m_1(\mathbb R^2)$, and the problem page records the bound $f(n)\ge m_1(\mathbb R^2)\,n$ of Larman and Rogers, which this paper does not state; so each independent set the lemma accepts yields a lower bound for $f(n)/n$. The paper reports no value above Croft's $0.22936$, and the lemma neither improves the known estimates of $f(n)$ nor decides whether $f(n)\ge n/4$.