= Solution
Write $X=\{x_1,\ldots,x_s\}$. Since $X$ is not a <spherical point set>, there are real coefficients $\lambda_i$, not all zero, such that
$$
\sum_i\lambda_i=0,
\qquad
\sum_i\lambda_i x_i=0,
\qquad
b:=\sum_i\lambda_i\lVert x_i\rVert^2\ne0.
$$
Indeed, take a minimal nonspherical subset; its points are affinely dependent, and centering the proper spherical subset shows that the corresponding quadratic sum is nonzero. The three relations are invariant under <isometry>[isometries], and rescaling the $\lambda_i$ lets us assume $b=1/2$.
Choose $\delta<1/(2s)$ and color every $y$ by the $s$ intervals of length $\delta$ containing the <fractional part>[fractional parts] of $\lambda_i\lVert y\rVert^2$. This uses finitely many colors. If $y_1,\ldots,y_s$ were a monochromatic isometric copy of $X$, then each
$$
\lambda_i(\lVert y_i\rVert^2-\lVert y_s\rVert^2)
$$
would lie within $δ$ of an integer. Their sum is within $s\delta<1/2$ of an integer, but because $\sum_i\lambda_i=0$ it equals
$$
\sum_i\lambda_i\lVert y_i\rVert^2=b=\frac12,
$$
a contradiction. Hence $X$ is not a <Euclidean Ramsey set>.
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