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Source. Lemma 1, p. 6, 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, the definitions of Section 3 (p. 4), the coordinates of the added points (p. 5) and the account of the certificate (p. 6) were read clause by clause on the print. The rational certificate, which the paper places in its supplementary material, was not checked here. Nothing here is independently reviewed.

Statement

Setting (p. 4). A fractional coloring of a graph $G$ is a nonnegative weight $\gamma$ on its independent sets giving every vertex total weight at least $1$; write $\overline\gamma(S)$ for the total weight of the independent sets containing $S\subseteq V(G)$. The coloring is geometric when $\overline\gamma(S)=\overline\gamma(S')$ for all $S,S'\subseteq V(G)$ that are isometric as subsets of the plane, and $\chi_{gf}(G)$ is the least total weight of a geometric fractional coloring. Always $\chi_f(G)\le\chi_{gf}(G)$.

Setting (p. 5). With $\omega_1=1/2+i\sqrt3/2$ and $\omega_3=5/6+i\sqrt{11}/6$, the configuration $G_{27}$ lies in the Moser lattice $\{a+b\omega_1+c\omega_3+d\omega_1\omega_3:a,b,c,d\in\mathbb Z\}$, and $G_{29}$ is the unit-distance graph on $V(G_{27})\cup\{p,q\}$, where

$$p=3+\tfrac{17}{8}\omega_1-\tfrac78\omega_3+2\omega_1\omega_3 +\sqrt5\left(-\tfrac14+\tfrac18\omega_1-\tfrac18\omega_3+\tfrac14\omega_1\omega_3\right),$$ $$q=\tfrac{11}{4}+\tfrac{13}{8}\omega_1-\tfrac18\omega_3+2\omega_1\omega_3 +\tfrac{\eta}{8}\left(-\omega_1+\omega_3+\omega_1\omega_3\right), \qquad \eta=i\sqrt{\tfrac{415+79\sqrt{33}}{8}}.$$

Both $p$ and $q$ have degree $1$, each adjacent only to the vertex $v_3$ (p. 6).

Lemma 1 (p. 6). $\chi_{gf}(G_{29})>4.0007$.

Proof pointer

A computer certificate: a rational feasible solution of the dual of the linear program defining $\chi_{gf}(G_{29})$, given in the authors' supplementary material and verified as in the earlier paper of Matolcsi, Ruzsa, Varga and Zsámboki (p. 6). The paper does not claim the certificate is optimal and does not determine $\chi_{gf}(G_{29})$; that dual program has 16860 variables and 498168 constraints, and its exact solution was beyond the authors' computational resources (p. 6). $G_{27}$ lies in the Moser lattice, and by Dúcz's arXiv:2606.12325 no graph in that lattice has $\chi_{gf}>4$ (p. 5), so $G_{29}$ does not lie in it.

Bears on

• Problem 1070: the input to Theorem 1, which turns this bound into a finite unit-distance graph with independence ratio below $1/4$.