Determine which countable ordinals $\beta$ have the property that, if $\alpha ={\omega }^{{}^{\beta }}$, then in any red/blue colouring of the edges of $K_{\alpha }$ there is either a red $K_{\alpha }$ or a blue $K_3$.

gpt-erdos · GPT-5.2 Pro + Deep Research · unverified

Write the statement in arrow notation as [ \omega^\beta \to (\omega^\beta,3)^2, ] i.e. every red/blue colouring of ([\omega^\beta]^2) yields either a red-homogeneous set of order type (\omega^\beta) [[nomath]](a red $K_{\omega^\beta}$)[[/nomath]] or a blue-homogeneous set of size $3$ (a blue triangle).

llm-hunter · gpt pro 5.2 · unverified

1 LLM attack(s) recorded (gpt pro 5.2); unverified.