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Source: published paper, printed p. 219, Question 1.1 and the conventions of Sections 1–3.

Write $\ell_m=\{0,e_1,\ldots,(m-1)e_1\}$. The notation $\mathbb E^n\to(X,K)$ means that every red-blue coloring of $\mathbb R^n$ has a red congruent copy of $X$ or a blue congruent copy of $K$. Congruences include reflections. The colorings are arbitrary; no regularity hypothesis is imposed. A set is $t$-separated if distinct points have distance at least $t>0$. Throughout $n\ge1$, $\log$ means $\log_2$, and $\ln$ is the natural logarithm.

A finite $X\subset\mathbb R^d$ is Ramsey if, for every positive integer $r$, some dimension $n\ge d$ has a monochromatic congruent copy of $X$ in every coloring with at most $r$ colors. The quantitative $f$-Ramsey notion is specified on Theorem 3.1.

For $L>2$ let $\mathbb T_L^n=\mathbb R^n/L\mathbb Z^n$, with distance

$$d_L(p,q)=\min_{z\in\mathbb Z^n}|p-q+Lz|.$$

Choose a maximal $1/3$-separated finite set $P$ in this torus and choose its representatives in $[0,L)^n$. Let $\Gamma=P+L\mathbb Z^n$. The lifted closed Voronoi cell of $p\in\Gamma$ is

$$V_p=\{x:|x-p|\le |x-q|\text{ for every }q\in\Gamma\}.$$

Maximality implies that every point is within $1/3$ of a center, so $V_p\subset\overline B(p,1/3)$ and $\operatorname{diam}V_p\le2/3$. The packing bound on the linked Lemma 2.1 page also shows that the greedy construction of $P$ stops after finitely many choices.

Closed cells may overlap on their boundaries. Selected cells will be colored red including their boundaries. A separate deterministic label rule in the main proof handles counting; it does not change the coloring. A copy of a configuration always means an ordinary Euclidean copy in the lifted space, not an assertion that all its distances survive quotienting to the torus.

The source uses period $R$, where the configuration has diameter at most $R-1$. The reconstruction uses period $L=3R$ to separate all periodic Bernoulli neighborhoods. This is a compilation-supplied modification, explained on the periodic-construction and main-theorem pages.

Related proof pages. lemma 2 1.

Bears on. Problem 188, Problem 174, Problem 214.