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Source. Conjecture 1, p. 3, of Ákos Dúcz and Dániel Varga, A unit-distance graph in the plane with independence ratio below 1/4, arXiv:2606.28157v1 (26 June 2026), the version named on the source card. A preprint.

Read depth. Claims checked: the statement and the paragraph before it were read on p. 3 of the print.

Statement

Setting (pp. 1--3). $f(n)$ is the largest integer such that every set of $n$ points in the plane contains $f(n)$ points no two at distance $1$; $m_1(\mathbb R^2)$ is the supremum of the upper densities of measurable planar sets with no two points at distance $1$; and $\delta_{\mathrm{Croft}}=0.22936\ldots$ is the density of Croft's construction, which gives $m_1(\mathbb R^2)\ge\delta_{\mathrm{Croft}}$.

Conjecture 1 (p. 3). $f(n)/n=m_1(\mathbb R^2)+o(1)$, and in particular $m_1(\mathbb R^2)=\delta_{\mathrm{Croft}}$.

Scope

The authors' own conjecture, stated after the remark that estimating $f(n)/n$ accurately is still wide open (p. 3). It is distinct from Conjecture 1 of Matolcsi, Ruzsa, Varga and Zsámboki, which Corollary 1 falsifies. The paper proves nothing toward it.

Bears on

• Problem 1070: a conjectured asymptotic for $f(n)$, the estimate the problem asks for; it remains a conjecture.
• Problem 232: a conjectured value of $m_1$, the quantity the problem asks to estimate; it remains a conjecture.