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Statement

Setting (p. 4). $\chi_{f,0}(\mathbb R^2)$ is the supremum of the fractional chromatic number $\chi_f(G)$ over finite unit-distance graphs $G\subseteq\mathbb R^2$.

Conjecture 1 (p. 10, quoted). "$\chi_{f,0}(\mathbb R^2)=4$, and for all finite unit distance graphs $G\subseteq\mathbb R^2$ we have $\chi_f(G)<4$."

The first clause adds to Corollary 1 the upper bound $\chi_{f,0}(\mathbb R^2)\le4$. By Theorem 2 it is equivalent to $\alpha_1(\mathbb R^2)=\frac14$, the form the paper also conjectures on p. 5, alongside $\delta_{\mathrm{Croft}}=m_1(\mathbb R^2)$. Since $\chi_f(G)\,\alpha(G)/|G|\ge1$ for every finite graph (p. 4), the second clause implies that every finite planar unit-distance graph $G$ has $\alpha(G)>|G|/4$. The paper notes (p. 8) that, since $m_1(\mathbb R^2)<\frac14$ was proved earlier, the conjecture would make the finitary and the measurable independence ratios of the plane differ. As support it reports (p. 14) that all children of $G_{27}$ under its search's extension rules, and a few hundred grandchildren, also have geometric fractional chromatic number $4$.

Source. Máté Matolcsi, Imre Z. Ruzsa, Dániel Varga, Pál Zsámboki, The fractional chromatic number of the plane is at least 4, arXiv:2311.10069, read in the version dated March 28, 2025 identified on the source card, whose page numbers are used here: the conjecture on p. 10, its motivation on pp. 2, 5 and 7--8, the computational checks on p. 14.

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

Bears on

• Problem 1070: the second clause implies $f(n)>n/4$ for every $n\ge1$, a positive answer to the particular question $f(n)\ge n/4$; given Corollary 1, the first clause is equivalent to every finite planar unit-distance graph having independence ratio at least $\frac14$. A finite graph with independence ratio below $\frac14$ would contradict both clauses; the problem page records a pending claim of such a graph. The conjecture itself has no standing as a result.