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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 1 (p. 174, quoted). "Every admissible coloring of the plane has a red translate of every three point configuration. In fact, every admissible coloring of $\mathbb{R}^m$ has a red translate of every $n$ point configuration, where $n\le(1+o(1))(1.2)^n$ [sic]."

The print writes the exponent of $1.2$ as $n$. The proof derives the second sentence from the Frankl–Wilson lower bound on $\chi(\mathbb{R}^m)$ (reference [1]), which is exponential in the dimension, so the intended range reads $n\le(1+o(1))(1.2)^m$; the paper does not state this correction itself.

Proof pointer

Section 2 (p. 174). Proposition 1 (p. 174): if some admissible coloring of $\mathbb{R}^m$ forbids red translates of an $n$-point configuration $A$, then $\chi(\mathbb{R}^m)\le n$. A point $p$ receives color $i$ when $p+a_i$ is blue (the least such $i$); every point is colored because no translate of $A$ is all red, and two points of color $i$ at distance one would give two blue points at distance one. Theorem 1 then follows from $\chi(\mathbb{R}^2)>3$ (cited from Hadwiger, Debrunner and Klee) and, in $\mathbb{R}^m$, from the Frankl–Wilson bound.

Read depth

Claims checked: Theorem 1, Proposition 1 and its proof were read clause by clause on the page images of the print. The chromatic-number bounds are cited by the paper, not proved there, and their sources were not read. Nothing here is independently reviewed.

Dependencies

None in the corpus. External inputs named by the paper: the lower bound $\chi(\mathbb{R}^2)>3$ (Hadwiger, Debrunner and Klee, Combinatorial Geometry in the Plane, 1964) and the Frankl–Wilson bound on $\chi(\mathbb{R}^m)$ (Combinatorica 4 (1981)).

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: a translate is a congruent copy, so Theorem 1 gives, in every planar coloring whose blue set avoids distance one, a red congruent copy of every three-point configuration. Problem 214 asks for the four vertices of a unit square; Theorem 1 covers three-point configurations only and does not decide that question. The paper recalls (p. 173) that Juhász had shown red congruent copies of every four-point configuration.