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Source. Published p. 232, the radius calculation after Lemma 3.12 and Lemma 3.13; the squared-length correction is explained below. (canonical PDF).

As printed (p. 232): for every integer $d\ge1$ there is $\mu=\mu(d+1)>0$ such that every $(\mu,\beta)$-regular simplex $T=\{t_1,\ldots,t_{d+1}\}$ with circumradius $\rho(T)=\rho^T$ is $\alpha$-hyper Ramsey for every $\alpha\ge\beta^2(d+1)^2-(\rho^T)^2$. Definition 3.1 needs $\alpha>0$, so only positive such $\alpha$ are meant. The printed derivation of this threshold treats a squared-distance bound as an edge length (see Source precision). The conclusion itself is not false: the repaired proof of Theorem 3.3 on this card, which does not use the printed threshold, makes every simplex $\alpha$-hyper Ramsey for every $\alpha>0$. The main proof needs an explicit threshold, and this card uses the corrected one below.

Corrected form proved here. Let $n=d+1\ge2$ and $0<\mu\le\mu_n=1/(n2^n)$. A $(\mu,\beta)$-regular simplex $T$ is $\alpha$-hyper-Ramsey whenever

$$\alpha\ge\beta n^2-\rho(T)^2.$$

The threshold on the right is strictly positive, and equality is allowed. This is the sufficient corrected version of the source's radius estimate.

Proof.

Lemma 3.12 places a congruent copy of $T$ in a box $P$ with

$$\rho(T)^2\le\rho(P)^2<\beta n^2.$$

The first inequality follows by projecting the box's containing center onto the affine span of the selected subset. In particular, the displayed threshold is positive.

For any allowed $\alpha$, set

$$R^2=\rho(T)^2+\alpha\ge\beta n^2>\rho(P)^2.$$

Theorem 3.2 makes $P$ hyper-Ramsey, so take its witnesses at positive squared slack $R^2-\rho(P)^2$. They lie exactly on $S(R,m)$ for every large $m$. A dense subset of one of these witnesses contains $P$ and therefore contains $T$. These same witnesses prove that $T$ is $\alpha$-hyper-Ramsey at its own squared slack $R^2-\rho(T)^2=\alpha$. The strict gap above $\rho(P)^2$ also handles equality in the allowed bound for $\alpha$.

Source precision.

Definition 3.11 bounds squared distances by $\beta(1+\mu)$. The source's following paragraph uses that quantity as an edge length and obtains $\rho(P)<\beta n$, leading to the printed budget $\beta^2n^2-\rho(T)^2$. With this definition of $\beta$, the actual edge length bound is $\sqrt{\beta(1+\mu)}$. The calculation in Lemma 3.12 gives the sufficient budget $\beta n^2-\rho(T)^2$ used here. This is a proved local repair of the estimate, not an author-issued erratum or a claim that the printed lemma's ultimate hyper-Ramsey conclusion is false. The repaired main proof chooses $\beta$ smaller accordingly.

No unrestricted hyper-Ramsey inheritance by subsets is asserted: the radius $R$ here is explicitly large enough for the whole containing box.

Bears on. #174.