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on Erdős #1172 · set theory / ramsey theory

The statement

Establish whether the following are true assuming the generalised continuum hypothesis:

\omega_3 \to (\omega_2,\omega_1+2)^2,

\omega_3\to (\omega_2+\omega_1,\omega_2+\omega)^2,

\omega_2\to (\omega_1^{\omega+2}+2, \omega_1+2)^2.

Establish whether the following is consistent with the generalised continuum hypothesis:

\omega_2\to (\omega_1+\omega)_2^2,

or even $\omega_2 \to (\xi)_2^2$ for all $\xi<\omega_2$ .

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