= Solution
On $[\kappa]^\omega$, declare $A\sim B$ when their <symmetric difference> is finite. Use the <well-ordering theorem> to choose one representative $R_{[A]}$ from each equivalence class, and color by
$$
\boxed{c(A)=|A\triangle R_{[A]}|\pmod2.}
$$
Removing one point from a countably infinite set does not change its equivalence class and reverses this parity. For any $H\subseteq\kappa$ of order type $\omega$ and $x\in H$, both $H$ and $H\setminus\{x\}$ belong to $[H]^\omega$ and receive different colors. There is no homogeneous $H$ of <order type> $\omega$. This proves the negative <infinite-arity partition relation>
$$
\boxed{\kappa\nrightarrow(\omega)^\omega_2.}
$$
The slashed arrow in the PDF is essential.
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