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Source. Kříž, Permutation groups in Euclidean Ramsey theory, published pp. 899–902, Sections 2.1–2.4 and Definition 3.1 (publisher PDF).

Configurations and colorings

A configuration is a finite subset of a Euclidean space. We use nonempty configurations in the proofs; the empty configuration, if admitted, satisfies all the Ramsey conclusions trivially. Dimensions are nonnegative integers, and a number of colors is a positive integer. Write $[m]=\{0,\ldots,m-1\}$, with $[0]=\varnothing$.

An isometrical embedding is an injective map preserving all pairwise Euclidean distances. An isometry of a finite configuration onto itself is a permutation of its points preserving distance. Its group of isometries is therefore finite. A group of ambient isometries acting on a configuration may be replaced by its finite induced permutation group. A group is soluble (or solvable) if its derived series, obtained by repeatedly taking the subgroup generated by commutators, reaches the identity subgroup after finitely many steps.

Let $E$ be an equivalence relation on $F$. The configuration is $E$-Ramsey if, for every integer $k\ge1$, there is a dimension $N$ such that every coloring $c:\mathbb R^N\to[k]$ admits an isometrical embedding $\phi:F\to\mathbb R^N$ with

$$xEy\quad\Longrightarrow\quad c(\phi(x))=c(\phi(y)).$$

Different $E$-classes may receive the same color. Ordinary Ramsey means $E=F\times F$. For an integer $s\ge1$, $s$-Ramsey means that the image of some such embedding uses at most $s$ colors. The partition in this last definition may initially depend on the coloring; Proposition 2.4.1 removes that dependence.

If $F_i$ carries $E_i$, the Euclidean product has squared distance

$$d((x_1,x_2),(y_1,y_2))^2 =d(x_1,y_1)^2+d(x_2,y_2)^2,$$

and $(x_1,x_2)(E_1\times E_2)(y_1,y_2)$ means $x_iE_iy_i$ for both coordinates. Powers $F^m$ and $E^m$ use the corresponding coordinatewise definitions.

Group actions and mergers

For a group $G$ acting on a finite set $F$, put

$$xE_Gy\quad\Longleftrightarrow\quad gx=y\text{ for some }g\in G.$$

A permutation $b$ respects $E$ if $xEy$ implies $b(x)E b(y)$. Because $b$ has finite order, its inverse also respects $E$, so it acts on $F/E$. If every element of $G$ respects $E$, define

$$\operatorname{St}_G(E)=\{g\in G:gxEx\text{ for every }x\in F\}, \qquad G/E=G/\operatorname{St}_G(E).$$

The stabilizer here is the kernel of the action on $F/E$, not the stabilizer of one point. The group $G/E$ is the induced permutation group of the equivalence classes.

Write $U(E;G)$ for the equivalence relation that joins all $E$-classes in each orbit of this quotient action. Explicitly,

$$x\,U(E;G)\,y\quad\Longleftrightarrow\quad gxEy\text{ for some }g\in G. \tag{1}$$

This is an equivalence relation because it is orbit equivalence on $F/E$, pulled back to $F$.

For $z\in F$, a permutation $b$ respecting $E$, and an integer $n\ge0$, put

$$A_n=\bigcup_{i=0}^{n-1}[b^iz]_E, \qquad U(E;z,b,n)=E\cup(A_n\times A_n). \tag{2}$$

Thus exactly the classes meeting $z,bz,\ldots,b^{n-1}z$ are merged. For $n=0$ or $1$ the relation is $E$. The merger need not be preserved by $b$ until an entire orbit of classes has been merged.

Source conventions

The printed norm convention on p. 900 writes $\|x\|=x\cdot x$. We use the ordinary Euclidean convention $\|x\|^2=x\cdot x$, consistent with the distance and scalar-product arguments throughout the paper. All inequalities defining $s$-Ramsey and the two-class results are inclusive: at most $s$ colors, at most two classes, or at most two orbits.

Related results. elementary closure properties, Proposition 2.4.1, and Theorem 4.1.

Bears on. Problem 174.