Which countable ordinals are partition ordinals: when is $\omega^\beta\to(\omega^\beta,3)^2$? (Erdős #592)

For a countable ordinal $\beta$ set $\alpha=\omega^{\beta}$. Say $\beta$ has the partition property if every red/blue colouring of the edges of the complete graph $K_\alpha$ on $\alpha$ vertices contains either a red $K_\alpha$ (a red-complete subgraph on an $\alpha$-sized vertex set) or a blue triangle $K_3$ — equivalently $\alpha\to(\alpha,3)^2$. Determine exactly which countable ordinals $\beta$ have this property. (Ordinals $\alpha$ with $\alpha\to(\alpha,3)^2$ are called partition ordinals.)

Acceptance. FULLY RESOLVES: a complete classification, with proof, of exactly which countable $\beta$ satisfy $\omega^\beta\to(\omega^\beta,3)^2$ — in particular settling the Galvin–Larson conjecture (that every additively indecomposable $\beta\geq3$ has the property) by proving it or exhibiting a counterexample; machine-checkable (Lean/Coq) preferred. ADVANCES: resolve the outstanding open case — $\beta=\omega^{\gamma}$ with $\gamma$ a sum of exactly three indecomposable ordinals — in either direction with full proof; or settle a single explicit new instance — $\beta=\omega^{\gamma}$ for a specific $\gamma$ that is a sum of exactly three indecomposable ordinals — not already covered by [Sp57], [Ch72], [Sc10] (the results stated in background), with full proof. Deliver the proof or formal development establishing the partition relation, or its failure, for the claimed ordinals.