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Statement

Definition (p. 283). Let $X_2$ be a set of two points at distance $1$. The chromatic number $\chi(\mathbb{E}^n)$ is the least $m$ such that $\mathbb{E}^n\not\xrightarrow{m}X_2$, that is, the least number of classes in a partition of $\mathbb{E}^n$ none of whose classes contains two points at distance $1$.

Bounds recorded (pp. 283–284). The chapter derives $4\le\chi(\mathbb{E}^2)\le7$ from the seven-point Moser graph, all of whose edges have length 1, and from a periodic seven-coloring of a tiling of the plane by regular hexagons of diameter $0.9$, and says these bounds "have remained unchanged for over 50 years" (p. 283). It reports O'Donnell's Theorem 11.1.5 (p. 284): for every $g>0$ there is a 4-chromatic unit distance graph in $\mathbb{E}^2$ with girth greater than $g$, which the author offers as evidence, in his opinion, that $\chi(\mathbb{E}^2)\ge5$.

Problem 11.1.6 (p. 284). Determine the exact value of $\chi(\mathbb{E}^2)$.

After the problem the chapter records (p. 284), with references, $(1.239+o(1))^n<\chi(\mathbb{E}^n)<(3+o(1))^n$ and $6\le\chi(\mathbb{E}^3)\le15$; Soifer's partition of the plane into seven classes, six with no two points at distance $1$ and the seventh with no two points at distance $1/\sqrt5$; Falconer's theorem that in every partition of the plane into four Lebesgue measurable sets one set contains two points at distance $1$; and the remark that the value of $\chi(\mathbb{E}^2)$ may depend on the axioms of set theory in use.

Scope

Problem 11.1.6 is an open problem as posed, not a result. The bounds are those the chapter reports for its 2017 edition; it indicates the two plane bounds through the Moser graph and the hexagon coloring and cites sources for the rest.

Source. R. L. Graham, Euclidean Ramsey theory, Chapter 11 of J. E. Goodman, J. O'Rourke and C. D. Tóth (eds.), Handbook of Discrete and Computational Geometry, 3rd edition, CRC Press, Boca Raton, FL, 2017; the definition and the bounds $4\le\chi(\mathbb{E}^2)\le7$ on p. 283, Theorem 11.1.5, Problem 11.1.6 and the remarks after it on p. 284. Pages are those printed on the edition named on the source card.

Read depth. Claims checked: the definition, the bounds, Theorem 11.1.5 and the problem were read clause by clause on the printed pages.

Bears on

• Problem 508: Problem 11.1.6 asks the same question, the chromatic number of the plane. The chapter records $4\le\chi(\mathbb{E}^2)\le7$; later results are recorded on the problem page, among them de Grey's lower bound of 2018, which the chapter predates.