Matolcsi 2025 fractional chromatic number plane is at

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conjecture_1: The paper conjectures that the finitary fractional chromatic number of the plane equals 4 while every finite planar unit-distance graph has fractional chromatic number below 4.

corollary_1: The fractional chromatic number of the unit-distance graph of the plane is at least 4, and the proof gives the same bound for its finitary version.

theorem_1: The supremum of the geometric fractional chromatic number over finite planar unit-distance graphs equals the supremum of their fractional chromatic number.

theorem_2: The supremum of the fractional chromatic number over finite planar unit-distance graphs equals the reciprocal of the infimum of their independence ratios.

theorem_3: The 27-vertex planar unit-distance graph G27 on the Moser lattice has geometric fractional chromatic number exactly 4, certified by a linear program and a rational dual witness.

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Máté Matolcsi, Imre Z. Ruzsa, Dániel Varga, Pál Zsámboki, The fractional chromatic number of the plane is at least 4. arXiv preprint (2025). arXiv:2311.10069. No notice is printed, and the file carries no arXiv stamp (it is dated March 28, 2025, matching the fourth version); the arXiv abstract page names arXiv's non-exclusive distribution license for the current version (https://arxiv.org/abs/2311.10069, read 2026-10-02), every other right reserved.

The paper proves the lower bound 4 for the fractional chromatic number of the unit distance graph of the plane (Corollary 1), without exhibiting any finite planar unit-distance graph of fractional chromatic number 4. The route is via the geometric fractional chromatic number: Theorem 1 shows the finitary geometric fractional chromatic number of the plane equals the finitary fractional chromatic number, using amenability of the group of planar Euclidean transformations to run a blow-up argument, and Theorem 3 exhibits an explicit 27-vertex planar unit-distance graph with geometric fractional chromatic number exactly 4. Theorem 2 shows the finitary fractional chromatic number equals the Hall ratio of the plane, so with Corollary 1 finite unit-distance graphs with independence ratio at most 1/4 + epsilon exist for every epsilon > 0; Conjecture 1 predicts that the value 1/4 itself is unattainable by a finite graph. For problem 508 this is the primary source for the lower bound 4 on the fractional chromatic number of the plane, obtained from Theorem 1 and the 27-vertex graph of Theorem 3; since the chromatic number is at least the fractional one, it gives only chi >= 4 for the chromatic number itself. For problem 1070, Theorem 2 with Corollary 1 gives finite graphs of independence ratio at most 1/4 + epsilon, hence, by disjoint copies (a step the paper does not write out), f(n) <= (1/4 + o(1))n, and Conjecture 1, which implies that every finite planar unit-distance graph has independence ratio above 1/4, is the statement that Dúcz and Varga's 2026 preprint claims to refute with a graph of independence ratio below 1/4.

Source: https://arxiv.org/abs/2311.10069.

Bears on.

• #508: the problem asks for the chromatic number of the plane. Corollary 1 proves chi_f(R^2) >= 4 for the fractional chromatic number; since chi >= chi_f, it gives only chi >= 4 for the chromatic number, weaker than the lower bounds recorded on the problem page.
• #1070: the problem asks for the order of f(n) and whether f(n) >= n/4. Theorem 2 with the finitary bound chi_{f,0}(R^2) >= 4 gives, for every epsilon > 0, a finite unit-distance graph with independence ratio at most 1/4 + epsilon (p. 7), from which f(n) <= (1/4 + o(1))n follows by disjoint copies, a step the paper does not write out; it does not decide whether f(n) >= n/4. Conjecture 1 implies f(n) > n/4 for every n; it is a conjecture, not a result.

Results.

• Theorem 1 (p. 5): chi_{gf,0}(R^2) = chi_{f,0}(R^2), the finitary geometric fractional chromatic number equals the finitary fractional chromatic number; proof by averaging over Følner sets of the countable solvable group generated by the isometries matching subsets of at least two vertices.
• Theorem 2 (p. 7): chi_{f,0}(R^2) = rho(R^2), the finitary Hall ratio, i.e. the supremum of |G|/alpha(G) over finite planar unit-distance graphs.
• Theorem 3 (p. 8): the 27-vertex graph G27 on the Moser lattice (Definitions 1 and 2, p. 8) has chi_gf(G27) = 4, certified by a linear program and a rational dual witness in the supplementary material.
• Corollary 1 (p. 10): chi_f(R^2) >= 4; the proof gives chi_{f,0}(R^2) >= 4, improving the previous published bound 3.8991 of Bellitto, Pêcher and Sédillot.
• Conjecture 1 (p. 10): chi_{f,0}(R^2) = 4, and chi_f(G) < 4 for every finite unit-distance graph G in the plane, so every such graph would have independence ratio above 1/4.

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