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Statement

Setting as in Theorem 1: $D_7$ is the unit-quadrance graph on $\mathbb F_7^2$.

Example 1 (p. 4). $\chi(\mathbb F_7^2)=\chi(D_7)=4$.

The paper gets $3\le\chi(D_7)\le4$ first: the upper bound from Theorem 1 ($(7+1)/2=4$), the lower bound from a cycle of length $7$ in $D_7$. It takes $a=5$ and $t=3$, for which $a^2+1=5$ and $-t^2+a^2+1=3$ are non-squares in $\mathbb F_7$, and displays the resulting 4-coloring as Table 1 (p. 4). The absence of a 3-coloring is stated as verified by computer, with no details given; a backtracking search run here also finds no proper 3-coloring of $D_7$.

Source. Le Anh Vinh, On chromatic number of unit-quadrance graphs (finite Euclidean graphs), arXiv:math/0510092v1 (2005), Example 1 and Table 1 on p. 4; the edition read is identified on the source card.

Read depth. Proof verified: Table 1 was checked here by computer to be a proper coloring of $D_7$ (under either reading of rows and columns as coordinates), and an exhaustive backtracking search here confirmed that $D_7$ has no proper 3-coloring. Nothing here is independently reviewed.

Proof pointer

Page 4: Theorem 1's construction with the stated $a$ and $t$, and an unspecified computer search for the lower bound.

Dependencies

Theorem 1, Lemmas 1 and 3 of the same paper.

Bears on

No Erdős problem in the corpus is linked to this result.