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Source. Published p. 221, Theorem 3.3; full proof in Section 4, pp. 232–234. (canonical PDF).

Every finite simplex $X$ is hyper-Ramsey. Precisely, for every $\alpha>0$ there are $c>1$, $0<\epsilon<1$ and an integer $m_0$ such that each $m\ge m_0$ has a finite nonempty witness

$$H_m\subseteq S(\sqrt{\rho(X)^2+\alpha},m),\qquad |H_m|<c^m,$$

and every $K\subseteq H_m$ with $|K|\ge(1-\epsilon)^m|H_m|$ contains a congruent copy of $X$. All essential same-paper deductions are proved in the linked pages. The exact deep external inputs remain those stated in Theorem 2.2, Lemma 2.3, and Theorem 3.2.

Proof.

Singletons were handled in definitions. Otherwise, let $X=\{x_1,\ldots,x_n\}$, $n=d+1\ge2$. Scaling and radius enlargement reduce the problem to $\rho(X)=1$ and an arbitrary squared slack $0<\alpha<1$: after proving that range, a larger slack follows by lifting, and scaling back multiplies squared slack by $\rho(X)^2$.

Let $e_{ij}=\|x_i-x_j\|^2$. The negative-type criterion and compactness of the unit zero-sum subspace give $\gamma>0$ such that

$$Q_e(\lambda)\le-\gamma \quad\text{if }\sum_i\lambda_i=0,\quad\sum_i\lambda_i^2=1.$$

Choose a positive number $\beta$ with

$$\beta<\min\left\{\frac{\gamma}{n^2},\frac{\alpha}{2n^2}\right\},$$

and define a new zero-diagonal array by $e'_{ij}=e_{ij}-\beta$ only for $i\ne j$. For every such unit zero-sum vector,

$$Q_{e'}(\lambda)=Q_e(\lambda)+\frac\beta2<0.$$

Theorem 2.1 therefore realizes $e'$ as the squared distances of a simplex $Z=\{z_1,\ldots,z_n\}$. In particular, its distinct pair distances are positive; this is a conclusion of the realization criterion. As $\beta\downarrow0$, circumradius_continuity gives $\rho(Z)\to1$. Decrease $\beta$ further, if needed, so that

$$\rho(Z)\le1+\alpha/8.$$

All preceding strict bounds persist under this decrease.

Set $\mu=1/(n2^n)$ and $\theta=\min\{\alpha,\beta\mu\}>0$. Lemma 3.9 produces a simplex $V=\{v_1,\ldots,v_n\}$ satisfying

$$\bigl|\|v_i-v_j\|^2-(e_{ij}-\beta)\bigr|\le\theta\quad(i\ne j),$$

which is $\alpha_V$-hyper-Ramsey for

$$\alpha_V=\rho(Z)^2(1+\theta/8)-\rho(V)^2>0.$$

Define the residual array $f$ by $f_{ii}=0$ and $f_{ij}=e_{ij}-\|v_i-v_j\|^2$ for $i\ne j$. It satisfies

$$\beta(1-\mu)\le\beta-\theta\le f_{ij} \le\beta+\theta\le\beta(1+\mu)\quad(i\ne j).$$

These inequalities alone are not being used as an unproved Euclidean realization assertion. The array form of lemma_3_12 constructs a simplex $T=\{t_1,\ldots,t_n\}$ with precisely these squared distances. Lemma 3.13 then gives its strictly positive admissible squared slack

$$\alpha_T=\beta n^2-\rho(T)^2>0.$$

By Lemma 3.4, $V*T$ has density witnesses in every sufficiently large dimension on the sphere whose squared radius is

$$\begin{aligned} R_0^2 &=\rho(V)^2+\alpha_V+\rho(T)^2+\alpha_T\\ &=\rho(Z)^2(1+\theta/8)+\beta n^2\\ &<(1+\alpha/8)^3+\alpha/2\le1+\alpha. \end{aligned}$$

For the last elementary estimate, $0<\alpha<1$ gives

$$(1+\alpha/8)^3-1 =\frac{3\alpha}{8}+\frac{3\alpha^2}{64}+\frac{\alpha^3}{512} \le\frac{217}{512}\alpha<\frac\alpha2.$$

The diagonal subset $\{(v_i,t_i):1\le i\le n\}$ of $V*T$ is congruent to $X$, because each squared pair distance is

$$\|v_i-v_j\|^2+\|t_i-t_j\|^2=e_{ij}.$$

Consequently those product witnesses force $X$ at their actual radius $R_0$. Lift them by one constant coordinate to radius $\sqrt{1+\alpha}$. As proved in definitions, this preserves all distances and the cardinality estimate and changes the exponential density parameter by at most a fixed factor. With the new dimension $m=N+1$, all sufficiently large integers $m$ occur. Thus they are the required $\alpha$-hyper-Ramsey witnesses for $X$. Since the positive squared slack was arbitrary after scaling and lifting, the theorem follows.

Source precision.

The source's Section 4 has the same contraction, approximation, near-regular residual and product-diagonal strategy. The corrected $O_n(\beta)$ radius budget from Lemma 3.13 requires $\beta<\alpha/(2n^2)$ instead of the printed $\beta<\sqrt{\alpha/(2n^2)}$. The smaller choice is compatible with all the other requirements and completes the method.

Off-diagonal contraction leaves the diagonal zero. The auxiliary regular simplex in Remark 4.1 has edge length $\sqrt\beta$, since $\beta$ is a squared distance. The existence of the residual simplex, continuity of its predecessor's circumradius, and the final dimension/exponent change are supplied explicitly above. The proof controls the actual containing sphere for the product before passing to its diagonal subset; it does not assume hyper-Ramsey inheritance at an arbitrary smaller intrinsic radius. These are compilation repairs and expansions, not an author-issued erratum.

Dependencies. theorem_2_1, lemma_3_4, lemma_3_9, lemma_3_12, lemma_3_13 and circumradius_continuity.

Bears on. #174.