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Statement

Setting (p. 173). A red–blue coloring of a Euclidean space is admissible if no two blue points are at distance one; an $n$-coloring is proper if no color class contains two points at distance one; $\chi(S)$ is the least $n$ for which $S$ has a proper $n$-coloring. An $n$-point configuration is a set $\{a_1,\ldots,a_n\}$ of $n$ points of $\mathbb{R}^m$, and its translates are the sets $A+v$.

Theorem 3 (p. 175, quoted). "Either every admissible coloring of the plane has a red translate of every four-point configuration, or there exists an admissible coloring of the plane and a seven-point configuration so that congruent copies of the seven point configuration are forbidden in the red."

The proof shows that the seven-point configuration in the second alternative can be taken to be a Moser spindle (Fig. 1, p. 175), placed so that its unit-distance pairs are at distance one.

Proof pointer

Section 2 (p. 175). Suppose an admissible coloring forbids red translates of a four-point configuration $\{a_1,a_2,a_3,a_4\}$ with $a_1=0$. The construction of Proposition 1 (see Theorem 1's page) gives a proper four-coloring of the plane whose first class is the blue set, so the red set is split into three classes each avoiding distance one. A Moser spindle needs more than three colors, so the red set contains no congruent copy of it.

Read depth

Claims checked: Theorem 3 and its proof were read clause by clause on the page images of the print. Nothing here is independently reviewed.

Dependencies

Proposition 1 of the same paper, stated on Theorem 1's page. External input named by the paper: the Moser spindle (Hadwiger, Debrunner and Klee, Combinatorial Geometry in the Plane, 1964).

Source. A. D. Szlam, Monochromatic translates of configurations in the plane, J. Combin. Theory Ser. A 93 (2001), 173--176, doi:10.1006/jcta.2000.3065; the edition read is named on the source card.

Bears on

• Problem 214: the second alternative would give an admissible planar coloring with no red congruent copy of a seven-point configuration. The paper does not decide which alternative holds, so Theorem 3 neither proves nor refutes any bound on the largest size of configuration forced in the red set.