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Statement

Setting (pp. 4, 7 and 10). $\chi_f(\mathbb R^2)$ and $\chi_{gf}(\mathbb R^2)$ are the suprema of the fractional chromatic number $\chi_f(G)$ and of the geometric fractional chromatic number $\chi_{gf}(G)$ (Definition 2, pp. 9-10) over finite unit-distance graphs $G$ in the plane; $\alpha_1(\mathbb R^2)$ is the infimum of $\alpha(G)/|G|$ over the same graphs (p. 10).

Conjecture (p. 10, unnumbered). The paper conjectures that "$\chi_f(\mathbb{R}^2)=\chi_{gf}(\mathbb{R}^2)=4$".

It rests on numerical evidence (p. 10): the authors' search found no finite planar unit-distance graph with $\chi_{gf}(G)>4$, the largest value found being $3.9954$. In the remark that precedes it (p. 10) the paper states, without proof and deferring the details to a follow-up publication, that $\alpha_1(\mathbb R^2)=1/\chi_f(\mathbb R^2)$ and $\chi_f(\mathbb R^2)=\chi_{gf}(\mathbb R^2)$, and says that the conjecture would then give $\alpha_1(\mathbb R^2)=\frac14$ alongside $m_1(\mathbb R^2)\le0.247$, so that the measurable and non-measurable independence ratios of the plane would differ.

Source. The conjecture and the remark before it, p. 10, of Gergely Ambrus, Adrián Csiszárik, Máté Matolcsi, Dániel Varga and Pál Zsámboki, The density of planar sets avoiding unit distances, Math. Program. 207 (2024), 303-327, arXiv:2207.14179; page numbers are those of arXiv:2207.14179v3, the edition named on the source card.

Read depth. Claims checked: the statement and the remark were read clause by clause on the printed page. It is a conjecture; the paper offers no proof.

Bears on

• Problem 1070: the paper does not mention the problem. The part $\chi_f(\mathbb R^2)\le4$ would give $\chi_f(G)\le4$ for every finite planar unit-distance graph $G$, hence $\alpha(G)\ge|G|/\chi_f(G)\ge|G|/4$, and so $f(n)\ge n/4$ for every $n$, a positive answer to the particular question (an observation of this page). A finite planar unit-distance graph with independence ratio below $\frac14$, or with $\chi_{gf}(G)>4$, would contradict the conjecture; the problem page records a pending claim of such a graph. The conjecture itself has no standing as a result.