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Statement

Definition (p. 3). $X_2$ is the set of two points at distance $1$. The chromatic number $\chi(\mathbb E^n)$ is the least $m$ such that $\mathbb E^n$ is not $m$-Ramsey for $X_2$, that is, the least $m$ for which some partition of $\mathbb E^n$ into $m$ classes has no class containing two points at distance $1$.

Bounds in the plane (p. 3). $4\le\chi(\mathbb E^2)\le7$. The lower bound comes from the seven-point Moser graph, all of whose edges have length $1$, which shows $\mathbb E^2\xrightarrow{3}X_2$; the upper bound from a periodic seven-colouring of a tiling of the plane by regular hexagons of diameter $0.9$. The chapter remarks that these bounds had stood unchanged for over fifty years.

Problem 11.1.6 (p. 4, quoted). "Determine the exact value of $\chi(\mathbb E^2)$."

Other dimensions (p. 4). The chapter reports, citing [FW81] (Frankl and Wilson) and [CFG91] (Croft, Falconer and Guy),

$$(6/5+o(1))^n<\chi(\mathbb E^n)<(3+o(1))^n,$$

and $6\le\chi(\mathbb E^3)\le15$, the lower bound due to Nechushtan [Nech00] and the upper bound to Radoičić and Tóth [RT02]. It also records Soifer's partition of the plane into seven classes $C_1,\ldots,C_7$ in which $C_1,\ldots,C_6$ contain no two points at distance $1$ and $C_7$ contains no two points at distance $1/\sqrt5$ [Soi92].

Source. R. L. Graham, Euclidean Ramsey theory, Chapter 11 of the Handbook of Discrete and Computational Geometry, 2nd edition, CRC Press (2004), read in the preprint of the chapter identified on the source card, whose own page numbers are cited: the definition and the planar bounds on p. 3, Problem 11.1.6 and the bounds in other dimensions on p. 4.

Read depth. Claims checked: the definition, the problem and each bound were read clause by clause on the page images of the preprint. The chapter sketches only the sources of the planar bounds and proves nothing; the cited papers were not read here. Nothing here is independently reviewed.

Proof pointer

No proof is printed beyond the pointers above: the Moser graph (Figure 11.1.1) for $\chi(\mathbb E^2)\ge4$, a hexagonal seven-colouring for $\chi(\mathbb E^2)\le7$, and the cited papers for the other bounds.

Dependencies

Theorem 11.1.5, which the chapter offers as evidence, in the author's opinion, that $\chi(\mathbb E^2)\ge5$.

Bears on

• Problem 508: the chapter's Problem 11.1.6 is the problem's question, and it reports $4\le\chi(\mathbb E^2)\le7$ as the bounds known to it.
• Problem 704: the chapter reports $(6/5+o(1))^n<\chi(\mathbb E^n)<(3+o(1))^n$, an exponential lower and upper bound on the chromatic number of the unit distance graph of $\mathbb E^n$. It does not address whether $\lim\chi(\mathbb E^n)^{1/n}$ exists.