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Source. Moore, arXiv:2608.09649v1, p. 2, Theorem 2.2 (canonical PDF). The primary input is P. Frankl and V. Rödl, A partition property of simplices in Euclidean space, Journal of the American Mathematical Society 3 (1990), 1–7, Theorem 5.1 on p. 5 and Definition 2.1 on p. 2 (published PDF on Frankl's institutional page, DOI).

Statement. Moore states the theorem as "Every finite affinely independent Euclidean configuration is Ramsey" (p. 2). Explicitly, if $S$ is such a configuration and $r\ge1$ is an integer, there is a dimension $N$ such that every $r$-coloring of $\mathbb R^N$ contains a monochromatic congruent copy of $S$. The singleton case is included.

Exact external input. Frankl–Rödl prove the stronger statement that an affinely independent set of $m$ points, identified with a subset of $\mathbb R^{m-1}$, is super-Ramsey. In their definition, for some $\epsilon>0$ and every sufficiently large $n$, a finite set $V_n\subseteq\mathbb R^n$ has the property that every subset avoiding the configuration has size less than $|V_n|/(1+\epsilon)^n$; their definition also gives an exponential upper bound for $|V_n|$. Taking $(1+\epsilon)^n>r$, one color class in $V_n$ has size at least $|V_n|/r$ and therefore contains the configuration. This explains the ordinary Ramsey form used here.

Proof scope. The complete Frankl–Rödl same-paper chain is compiled in Theorem 5.1, and its ordinary-color consequence is recorded separately. The exact external inputs to that earlier proof remain stated there. This page links the consequence used to make the auxiliary simplex Ramsey in Moore's Theorem 1.2.

Bears on. #174.