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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 2 (p. 174, quoted). "There exists a seven-point configuration and an admissible red-blue coloring of the plane so that the seven-point configuration is forbidden in the red set."

Here "forbidden in the red set" refers to translates, as the abstract (p. 173) and the proof make explicit: no translate of the configuration lies entirely in the red set.

Proof pointer

Section 2 (p. 175). The theorem follows by applying Proposition 2 to Isbell's hexagonal coloring of the plane, cited from Hadwiger, Debrunner and Klee. The paper states only this deduction; that Isbell's coloring is a regular proper coloring with seven classes is left implicit. Proposition 2 then gives an admissible two-coloring and a configuration made of the translation vectors of the classes, none of whose translates is all red.

Read depth

Claims checked: Theorem 2 and the deduction from Proposition 2 were read clause by clause on the page images of the print. The paper gives no further detail of Isbell's coloring, and the cited source was not read. Nothing here is independently reviewed.

Dependencies

Proposition 2 of the same paper. External input named by the paper: Isbell's hexagonal coloring (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 problem and its related threshold concern congruent copies. Theorem 2 forbids only red translates of its seven-point configuration, so it gives no admissible coloring avoiding red congruent copies of that configuration and does not bear on the unit-square question directly.