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Statement

Setting (p. 125). $\chi(n)$ is the least number of classes in a partition of $\mathbb{E}^n$ such that no class contains two points at mutual distance $1$.

Section 6 reports, without proof of its own, the bounds then known.

• The plane (p. 125, display (4)). $4\le\chi(2)\le7$. The print writes the display as $4\le\chi(n)\le7$; the surrounding sentence concerns $\mathbb{E}^2$ and attributes the question and these bounds to Nelson in 1950, citing Soifer's historical essay. The lower bound comes from the seven-point Moser graph (Fig. 2, p. 125), whose edges join points at unit distance; the upper bound comes from a 7-coloring of a tiling of the plane by regular hexagons of diameter $1-\varepsilon$. The paper adds that the bounds in (4) had not moved in 40 years.
• General $n$ (p. 126). $(1+o(1))(1.2)^n<\chi(n)<(3+o(1))^n$, citing Frankl and Wilson, Combinatorica 1 (1981); the lower bound is attributed to them and to one of their set intersection theorems.

Read depth

Claims checked: the definition, both displays and their attributions were read on the print. The section proves nothing itself; the bounds are the cited authors' results.

Dependencies

None in the corpus.

Source. R. L. Graham, Recent trends in Euclidean Ramsey theory, Discrete Math. 136 (1994), 119--127, doi:10.1016/0012-365X(94)00110-5; the edition read is named on the source card.

Bears on

• Problem 508: the problem asks for the chromatic number of the plane. The paper reports the bounds 4 and 7 as the best known in 1993 and proves no new bound.
• Problem 704: the problem asks for estimates of the chromatic number of the unit distance graph of $\mathbb{R}^n$, whether it grows exponentially, and whether its $n$-th root converges. The paper reports a lower bound $(1+o(1))(1.2)^n$ and an upper bound $(3+o(1))^n$, both from the cited literature; it does not discuss the limit.
• Problem 214: background only. A class of the hexagonal 7-coloring is a planar set with no two points at distance 1, the kind of set the problem starts from; the paper does not discuss the problem's question about unit squares in the complement of such a set.