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Source. Kenneth Moore, A pyramid with a Ramsey base is Ramsey, arXiv:2608.09649v1, Theorem 1.2 on p. 2, proof on pp. 3–5 (canonical PDF).

Statement. If $B$ is a finite Ramsey set in a Euclidean space and $z\notin\operatorname{aff}(B)$, then $X=B\cup\{z\}$ is Ramsey. Thus for every positive integer $r$ there is a dimension $N$ such that every $r$-coloring of $\mathbb R^N$ contains a monochromatic congruent copy of $X$. There is no transitivity or convex-projection assumption on the base or apex. For an empty base, if that convention is allowed, the conclusion is the elementary singleton case.

Inputs. The classical product theorem and Frankl–Rödl simplex theorem are exact external inputs. The finite-witness lemma has a complete proof relative to the stated Rado selection principle.

Complete relative proof. Assume $B\ne\varnothing$. Work up to congruence in $\operatorname{aff}(X)$. If $d=\dim\operatorname{aff}(B)$, orthogonal projection onto the base and a choice of unit normal identify

$$B\subseteq\mathbb R^d,\qquad z=(u,h)\in\mathbb R^{d+1}, \qquad h>0.$$

Here $B$ is embedded as $B\times\{0\}$, $u$ is unrestricted in $\mathbb R^d$, and for every $b\in B$,

$$\|(b,0)-z\|^2=\|b-u\|^2+h^2. \tag{1}$$

We induct on the number of colors. For one color, any Euclidean space containing a copy of $X$ suffices. Suppose $r\ge2$ and the assertion holds for $r-1$. Some dimension $d_2$ then satisfies $\mathbb R^{d_2}\to_{r-1}X$. Lemma 2.3 supplies a finite set of distinct points

$$C=\{c_1,\ldots,c_m\}\subseteq\mathbb R^{d_2}, \qquad C\to_{r-1}X.$$

In particular $m\ge1$. Let $e_1,\ldots,e_m$ be the standard orthonormal basis of $\mathbb R^m$ and set

$$s_i=(c_i,he_i),\qquad S=\{s_1,\ldots,s_m\}\subseteq\mathbb R^{d_2+m}.$$

If $\sum_i t_i=0$ and $\sum_i t_i s_i=0$, the last $m$ coordinates give $ht_i=0$ for every $i$. Since $h>0$, all $t_i$ vanish. Thus $S$ is affinely independent, and the simplex theorem makes $S$ Ramsey. The product theorem consequently makes

$$P=B\times S =\{(b,c_i,he_i):b\in B,\ 1\le i\le m\} \subseteq\mathbb R^{d+d_2+m}$$

Ramsey.

For each $i$, let

$$P_i=\{(b,c_i,he_i):b\in B\},\qquad a_i=(u,c_i,0),\qquad A=\{a_1,\ldots,a_m\}.$$

Within each $P_i$, distances equal the corresponding base distances. Moreover,

$$\|(b,c_i,he_i)-a_i\|^2=\|b-u\|^2+h^2. \tag{2}$$

The point $a_i$ is distinct from every point of $P_i$, since $he_i\ne0$. Equations (1)–(2) show that $P_i\cup\{a_i\}$ is congruent to $X$. Also $\|a_i-a_j\|=\|c_i-c_j\|$, so $A$ is congruent to $C$.

Choose $n_0$ with $\mathbb R^{n_0}\to_r P$ and put $N=\max(n_0,d+d_2+m)$. Embed $P$ and $A$ in $\mathbb R^N$ by adding zero coordinates. Restricting a coloring to a coordinate copy of $\mathbb R^{n_0}$ shows that every $r$-coloring of $\mathbb R^N$ contains a monochromatic copy $P'$ of $P$. Call its color red and let $\phi:P\to P'$ be a congruence.

We need to transport all the auxiliary apices by the same ambient isometry. To justify this, fix $p_0\in P$. Equality of pairwise distances implies, by polarization,

$$\langle p-p_0,q-p_0\rangle =\langle\phi(p)-\phi(p_0),\phi(q)-\phi(p_0)\rangle \qquad(p,q\in P).$$

Thus the rule $p-p_0\mapsto\phi(p)-\phi(p_0)$ extends linearly to an inner-product-preserving map between the spans of these vectors: a vanishing linear combination has squared norm zero after applying the rule as well. Extend orthonormal bases of the two spans to orthonormal bases of $\mathbb R^N$. Mapping the extra basis vectors correspondingly gives an orthogonal map $Q$ on $\mathbb R^N$. The ambient isometry

$$\widetilde\phi(v)=\phi(p_0)+Q(v-p_0)$$

extends $\phi$. Put $a_i'=\widetilde\phi(a_i)$.

If one $a_i'$ is red, then $\widetilde\phi(P_i\cup\{a_i\})$ is a red copy of $X$. Otherwise the set $\{a_1',\ldots,a_m'\}$ uses at most $r-1$ colors. It is congruent to $C$, so the defining property of $C$ again gives a monochromatic copy of $X$. This proves the induction step and hence the theorem. $\square$

Source precision. The ambient-embedding sentence on source p. 4 names $P$ and $C$; the apices subsequently transported lie in $A$. The proof above explicitly embeds $P$ and $A$ and supplies the ambient-isometry extension argument. These are explanatory expansions of the construction, not a new theorem or an author-issued erratum. No numerical dimension estimate or formal verification is claimed.

Relation to the later source. Mirabi's Theorem 1.1 has the same conclusion, with a different proof using equivalence relations and a cyclic product construction.

Bears on. #174.