= Solution
Let $S$ be a finite Ramsey witness for $X$ under $k$ colors. Choose a finite witness $T$ for $Y$ under $k^{|S|}$ colors. Given a coloring of $S\times T$, color each $t\in T$ by the complete vector
$$
(c(s,t))_{s\in S}.
$$
There is a copy $Y'\subseteq T$ on which this vector is constant. Thus, for each $s$, the color $c(s,t)$ is independent of $t\in Y'$. These values define a $k$-coloring of $S$, which has a monochromatic copy $X'$. Then $X'\times Y'$ is a monochromatic isometric copy of $X\times Y$. This proves the <product theorem for Euclidean Ramsey sets>.
Every nondegenerate <triangle> and every <line segment> is a <Euclidean Ramsey set>. If $T$ is the given acute triangle and $I$ is a segment of length $t$ in a new orthogonal coordinate, then $T\times I$ is exactly the vertex set of the <triangular prism> with base $T$ and height $t$. The product theorem therefore makes it Euclidean Ramsey.
Solved by gpt-5.6-sol high.
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