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Source. Daniel W. Cranston and Landon Rabern, The fractional chromatic number of the plane, arXiv:1501.01647 (2015); later Combinatorica 37 (2017), 837–861, doi:10.1007/s00493-016-3380-3. Lemma 1 on p. 8 of arXiv v1 (7 January 2015), the edition named on the source card; the journal's labels and pagination may differ.

Notation (p. 6). Fix a vertex $v$ of the unit triangular lattice. $G_d$ is the graph made of the lattice vertices within distance $d$ of $v$ (the core) together with every Moser spindle attached to the core in the three directions of Section 2; $C_d$ is the subgraph of $G_d$ induced by the core vertices. The eight tiles $T1,\dots,T8$ are drawn in Figure 5 (p. 9), up to reflection and rotation.

Statement

Lemma 1 (p. 8). The paper states:

Let $I$ denote a maximal independent subset in $C_d$. There exists a set $\mathcal T$ of 8 finite tiles (shown in Figure 5), independent of $d$ and $I$, such that $C_d$ can be tiled with tiles from $\mathcal T$ where each corner of each tile is a vertex of $I$ and no vertex of $I$ lies in the interior of any tile. In this tiling, each face of $C_d$ is covered by exactly one or two tiles. (We do allow tiles to extend past the boundary of $G_d$, though this allowance could be removed by adding more tiles to $\mathcal T$.)

Read depth. Claims checked: the statement and the definitions it uses were read clause by clause on the arXiv v1 PDF. The case analysis of the proof was read for structure; nothing here is independently reviewed.

Proof pointer

The proof runs over pp. 8–11, in Section 3.2 (pp. 8–12). Join two vertices of $I$ by a segment exactly when their Euclidean distance is less than $3$, and delete every pair of segments that cross; the faces of the resulting plane graph are the tiles. A face containing a lattice edge at one of its corners in its interior is identified as one of $T1$, $T3$–$T8$ by a case analysis on which vertices near that corner lie in $I$ (Figure 6, p. 10), using only that $I$ is independent and maximal. A face containing no lattice edge in its interior has every boundary segment of length $2$ and corner angles $\pi/3$, so it is $T2$. Figure 7 (p. 11) shows an example tiling.

Uses within this source

The discharging proof of Theorem 2 averages the final weight of the core vertices tile by tile over this tiling (pp. 12–18).

Bears on

• Problem 508: only through Theorem 2, a lower bound on the fractional chromatic number of the plane; the lemma itself is a statement about the triangular lattice and says nothing about the chromatic number.