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Source. Published pp. 215–217, 220–221 and 231: Definition 1.1 on p. 215, Definition 1.2 and the circumradius on p. 216, Definitions 1.4 and 1.5 on p. 217, Definition 3.1 on pp. 220–221, the product on p. 221 and Definition 3.11 on p. 231. (canonical PDF).

Write $S(R,m)=\{x\in\mathbb R^m:\|x\|=R\}$; $m$ is the ambient dimension. The paper defines the circumradius $\rho(X)$ of a spherical set as "the radius of the smallest sphere containing $X$" (p. 216), a sphere on which $X$ lies. That sphere is the one whose center lies in $\operatorname{aff}X$ (circumradius_continuity); it is not the radius of a smallest enclosing ball.

Definition 1.1. $X\subseteq\mathbb R^d$ is Ramsey if for every $\chi\ge2$ there is an integer $n=n(X,\chi)$ such that every $\chi$-colouring of $\mathbb R^n$ has a monochromatic subset congruent to $X$.

Definition 1.2. $X\subseteq\mathbb R^d$ is sphere Ramsey if for every $\chi\ge2$ there are an integer $n=n(X,\chi)$ and a real $\varrho=\varrho(X,\chi)>0$ such that every $\chi$-colouring of $S(\varrho,n)$ has a monochromatic subset of that sphere congruent to $X$.

Definition 1.4. $X\subseteq\mathbb R^d$ is exponentially Ramsey if there is a real $\sigma=\sigma(X)>0$ such that for every integer $n\ge d$ and every $\chi$-colouring of $\mathbb R^n$ with $\chi\le(1+\sigma)^n$ some monochromatic subset is congruent to $X$.

Definition 1.5. $X\subseteq\mathbb R^d$ with $\rho(X)=\rho$ is strong Ramsey if for every real $\delta>0$ there is a real $\sigma=\sigma(X)>0$ such that for every integer $n\ge d$ and every $\chi$-colouring of $S(\rho+\delta,n)$ with $\chi\le(1+\sigma)^n$ some monochromatic subset of that sphere is congruent to $X$. The eventual reading used in this compilation, for all sufficiently large $n$ with $\sigma$ depending on $X$ and $\delta$, is explained under Source precision.

Definition 3.1. For a real $\alpha>0$, $X\subseteq\mathbb R^d$ with $\rho(X)=\rho$ is $\alpha$-hyper Ramsey if there are reals $c=c(X,\alpha)$ and $\epsilon=\epsilon(X,\alpha)>0$ and an integer $m_0=m_0(\alpha)$ such that every $m\ge m_0$ has a finite subset $\mathcal H(m)\subseteq\mathbb R^m$ with

$$\text{(i) }\mathcal H(m)\subseteq S(\sqrt{\rho^2+\alpha},m),\qquad \text{(ii) }|\mathcal H(m)|<c^m,$$

and (iii) every $\mathcal K\subseteq\mathcal H(m)$ with $|\mathcal K|\ge(1-\epsilon)^m|\mathcal H(m)|$ contains a subset congruent to $X$. It is hyper Ramsey if it is $\alpha$-hyper Ramsey for every real $\alpha>0$. Since (iii) applied to $\mathcal K=\mathcal H(m)$ forces $\mathcal H(m)$ to be nonempty, (ii) needs $c>1$; the pages of this card take $c>1$ and $0<\epsilon<1$, which loses nothing. Equivalently, every $X$-free subset of $\mathcal H(m)$ has relative size strictly less than $(1-\epsilon)^m$. The printed $m_0=m_0(\alpha)$ is read as also depending on $X$.

Product (p. 221). For $X\subseteq\mathbb R^n$ and $Y\subseteq\mathbb R^m$, $X*Y=\{x*y:x\in X,\ y\in Y\}$, where $x*y$ is the concatenation $(x_1,\ldots,x_n,y_1,\ldots,y_m)$.

Definition 3.11. For reals $1\ge\mu\ge0$ and $\beta>0$, a simplex $T=\{t_1,\ldots,t_{d+1}\}$ is $(\mu,\beta)$-regular if $\beta(1-\mu)\le\|t_i-t_j\|^2\le\beta(1+\mu)$ for every $1\le i<j\le d+1$. Thus $\beta$ has the units of a squared length.

The elementary transfers below will be used with their exact radii.

Proof.

A singleton is hyper-Ramsey: in each dimension use one point on the specified sphere, any $c>1$ and any $0<\epsilon<1$. A subset of this witness meeting the positive density threshold is nonempty.

If witnesses on a fixed sphere $S(R,m)$ are moved by

$$x\longmapsto (x,\sqrt{R'^2-R^2})\in\mathbb R^{m+1},\qquad R'\ge R,$$

they lie exactly on $S(R',m+1)$, with all distances and cardinalities unchanged. If their avoiding density was less than $(1-\epsilon)^m$, put $1-\epsilon'=\sqrt{1-\epsilon}$. For $N=m+1\ge2$,

$$(1-\epsilon)^m\le(1-\epsilon)^{N/2}=(1-\epsilon')^N.$$

This proves radius enlargement in every sufficiently large dimension, including the equality case $R'=R$. Zero-coordinate padding alone preserves a fixed radius and distances. Scaling all coordinates by $t>0$ changes $\rho$ to $t\rho$ and squared slack $\alpha$ to $t^2\alpha$.

If a witness forces $B$, it also forces any nonempty $A\subseteq B$ on that same sphere. The squared slack for $A$ is then $R^2-\rho(A)^2$, not the squared slack originally assigned to $B$. In particular, this does not prove that $A$ is hyper-Ramsey at every arbitrarily small slack above its own intrinsic radius.

Hyper-Ramsey implies strong Ramsey. Given $\delta>0$, take $\alpha=(\rho+\delta)^2-\rho^2>0$ and the corresponding witnesses. If $q\le(1-\epsilon)^{-m}$, one color class has density at least $1/q\ge(1-\epsilon)^m$, including equality, and contains $X$. Thus one may take $1+\sigma=(1-\epsilon)^{-1}$ in the eventual assertion.

For completeness, an eventual strong assertion can be extended to every $m\ge d+1$, where $d=\dim\operatorname{aff}X$, by decreasing $\sigma$: choose it so that $(1+\sigma)^m<2$ in the finitely many earlier dimensions. Only the one-color case then remains, and $X$ fits on every sphere of radius $\rho+\delta$ in those dimensions by the one-coordinate lift. If only $q\ge2$ is quantified, the source's lower bound $m\ge d$ can be used instead, since the exceptional initial dimensions are vacuous.

Source precision.

Definition 1.5 prints $\sigma=\sigma(X)$ after quantifying $\delta$; the proof supplies dependence on $\delta$ as well. Its all-$m\ge d$ wording must be read with the earlier $q\ge2$ convention: a full $d$-simplex cannot be placed on a strictly larger sphere in $\mathbb R^d$ even with one color. The eventual formulation above and the explicit $m\ge d+1$ extension remove this ambiguity. The distinction between a containing sphere and an enclosing ball is essential.

Dependencies. The intrinsic-radius facts are proved in circumradius_continuity.

Bears on. #174.