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Source. Published p. 217, Theorem 1.6, deduced from Theorem 3.3 on p. 221. (canonical PDF).

As printed, Theorem 1.6 reads "Every simplex is strong Ramsey." (p. 217). By Definition 1.5 (p. 217), a set $X\subseteq\mathbb R^d$ with circumradius $\rho(X)=\rho$ is strong Ramsey when for every real $\delta>0$ there is a positive real $\sigma=\sigma(X)$ 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 the sphere is congruent to $X$.

Form proved here. Let $X$ be a finite simplex of intrinsic circumradius $\rho$ and let $\delta>0$. There are $\sigma>0$ and an integer $m_0$ such that, for every integer $m\ge m_0$ and every positive integer

$$q\le(1+\sigma)^m,$$

every $q$-coloring of $S(\rho+\delta,m)$ has a monochromatic congruent copy of $X$. One may decrease $\sigma$ to obtain the assertion for all $m\ge\dim\operatorname{aff}X+1$. No measurability of the coloring is required. This is the printed theorem read with the qualifications on the dimension range and on the dependence of $\sigma$ recorded under Source precision below.

Proof.

Use Theorem 3.3 with squared slack $\alpha=(\rho+\delta)^2-\rho^2>0$. For every large $m$, its finite witness $H_m$ lies on $S(\rho+\delta,m)$ and forces $X$ in every subset of relative size at least $(1-\epsilon)^m$. Set $1+\sigma=(1-\epsilon)^{-1}$. Under any allowed coloring, a largest color class on $H_m$ has relative size at least

$$\frac1q\ge(1+\sigma)^{-m}=(1-\epsilon)^m.$$

The weak density endpoint therefore gives a monochromatic copy even when the integer color bound is attained. This finite pigeonhole argument uses no regularity or measurability assumption. The earlier-dimension extension is the explicit small-$\sigma$ argument in definitions: only the one-color case remains in those finitely many dimensions, and the required sphere embedding exists once the dimension is at least the affine dimension plus one.

Source precision.

The printed definition writes $\sigma=\sigma(X)$ although $\sigma$ is chosen after $\delta$, and it quantifies every $n\ge d$ together with every $\chi\le(1+\sigma)^n$, which includes $\chi=1$; a full $d$-simplex lies on no sphere of radius larger than $\rho$ in $\mathbb R^d$, so the case $n=d$ cannot hold as printed. Both points are qualified in definitions, and the form above avoids them. This conclusion gives positive radius slack; it does not assert forcing on the intrinsic circumsphere itself.

Dependencies. theorem_3_3 and definitions.

Bears on. #174.