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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 $A+v$.

An $n$-coloring of $\mathbb{R}^m$ with classes $C_1,\ldots,C_n$ is regular (p. 174) if $C_i=C_1+v_i$ for some fixed vectors $v_1,\ldots,v_n$.

Proposition 2 (p. 174, quoted). "If $\mathbb{R}^m$ can be properly $n$-colored by a regular coloring, then there exists an admissible two-coloring of $\mathbb{R}^m$ and an $n$-point configuration $A$ so that translates of $A$ are forbidden in the red set."

The paper calls it a partial converse to Proposition 1, which bounds $\chi(\mathbb{R}^m)\le n$ whenever an admissible coloring forbids red translates of an $n$-point configuration (see Theorem 1's page).

Proof pointer

Section 2 (pp. 174--175). Normalize $v_1=0$, take $A=\{v_1,\ldots,v_n\}$, color $C_1$ blue and everything else red. The blue set avoids distance one because the coloring is proper. The proof shows that the points of any translate $p+A$ lie in distinct classes, using $C_a=C_1+v_a$ and the identity $(p+v_i-v_a)+v_j=(p+v_j-v_a)+v_i$; with $n$ classes and $n$ points, one point lies in $C_1$ and is blue.

Read depth

Claims checked: the definition of a regular coloring, Proposition 2 and its proof were read clause by clause on the page images of the print. Nothing here is independently reviewed.

Dependencies

None.

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 proposition is the construction behind Theorem 2; it produces colorings that forbid red translates, not red congruent copies, and the paper draws no conclusion from it about the unit-square question.