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Source. Published pp. 1–3 and 5, Definitions 1 and 6 and Section 2 (canonical PDF). The corresponding arXiv labels are Definitions 1.1 and 3.4.

For an integer $m\ge2$ and a real $r>0$, write

$$T_{m,r}=\{r(\cos(2\pi j/m),\sin(2\pi j/m)):0\le j<m\}.$$

A regular polygonal torus is a finite orthogonal Cartesian product $T=\prod_{a=1}^{s}T_{m_a,r_a}\subseteq\mathbb R^{2s}$, where $s\ge1$, $m_a\ge2$ and $r_a>0$. Only the vertices are included. The source permits $m=2$, interpreted as two antipodal points. If all $m_a=m$, the product is $m$-regular. If also all $r_a=r$, it is $(m,r)$-regular, denoted $T_{m,r}^{s}$. A common polygon order does not require a common radius.

An embedding preserves every Euclidean distance, without rescaling. For $\delta>0$, a $\delta$-embedding of a finite set $X$ into $T$ is an injection $f:X\to T$ satisfying

$$\left|\|f(x)-f(x')\|^2-\|x-x'\|^2\right|<\delta \qquad(x,x'\in X).$$

The error is in squared distance. Injectivity is an additional condition, not a consequence of an unrestricted additive error bound.

A simplex is a finite affinely independent configuration. For labeled points $x_1,\ldots,x_n$, affine independence means that $\sum_i c_i=0$ and $\sum_i c_ix_i=0$ force all $c_i=0$. Single points are included; the empty configuration has only vacuous embedding and Ramsey assertions. The substantive proofs take $n\ge2$.

A finite configuration $X$ is Ramsey if, for every integer $q\ge1$, there is a dimension $N=N(X,q)$ such that every coloring $\mathbb R^N\to[q]$ contains a monochromatic isometric copy of $X$. Here $[q]=\{1,\ldots,q\}$. The dimension may depend on the configuration and number of colors. This is an all-color assertion, stronger than a fixed two-color statement.

Every centered torus above lies on a sphere of radius $(\sum_a r_a^2)^{1/2}$. This radius belongs to the containing product. An embedded subset can have a smaller intrinsic circumradius, namely the radius about its equidistant center in its own affine hull. No definition here identifies those radii or prescribes the radius of a Ramsey witness.

The exact correction of nearly equal squared distances is in Lemma 4, and regular expansions are treated in the expansion identity.