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Source. Published p. 1, introduction, and p. 7, abstract, with Definition 2.1 and Theorem 5.1.

Statement. For the vertex set $A$ of each fixed nondegenerate simplex, there are $\epsilon>0$ and a threshold $n_0$ such that every coloring of $\mathbb R^n$, $n>n_0$, with an integer number $r\le(1+\epsilon)^n$ of colors contains a monochromatic copy of $A$. Consequently, the required dimension is $O_A(1+\log r)$, and every nondegenerate simplex is Ramsey.

The abstract (p. 7) states the color form without a threshold: for the vertex set $A$ of a nondegenerate simplex in $\mathbb R^d$ there is $\epsilon=\epsilon(A)>0$ such that every partition of $\mathbb R^n$ into fewer than $(1+\epsilon)^n$ parts has a part containing a set congruent to $A$. The introduction (p. 1) announces $n(r,B)=c(B)\log r$ for simplices, bricks and their products. The statement above is the form that follows from Theorem 5.1 and Definition 2.1, with the threshold made explicit.

Proof. Take the witnesses from theorem_5_1. Some color class in $X_n$ has size at least $|X_n|/r\ge|X_n|/(1+\epsilon)^n$. By the strict avoiding-set bound in Definition 2.1, that class contains $A$. For a given $r\ge1$, choose an integer $n>n_0$ with $n\ge\log r/\log(1+\epsilon)$. This gives the claimed dimension estimate and handles a fixed small number of colors by the threshold term. The singleton case is immediate independently of the estimate.

The source's introduction writes the logarithmic dimension relationship without rounding and threshold terms. These are supplied here. No explicit uniform constant over all simplex shapes is claimed.

Connections. This is the precise ordinary Ramsey input used in Moore's simplex input. It also supports the historical simplex examples in Conlon–Fox's outside-input record. It is stronger than ordinary finite-color forcing but does not classify all Ramsey configurations.

Bears on. #174: every nondegenerate simplex is Ramsey in the problem's sense, with dimension $O_A(1+\log r)$ for $r$ colors. This identifies one class of Ramsey sets; it is not a characterization of the Ramsey sets the problem asks for.