Erdős Problem #1172

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 true assuming the continuum hypothesis:\[\omega_2\to (\omega_1+\omega)_2^2.\]

#1172: [Va99,7.87]

set theory | ramsey theory

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A problem of Erdős and Hajnal. The Erdős-Rado partition theorem [ErRa56] states that\[(2^{\kappa})^+ \to (\kappa^++1)_\kappa^2\]for every infinite cardinal $\kappa$.

(The right-hand side of the first and final statements are missing from the truncated photocopy available of [Va99], and it is possible they have been filled in incorrectly.)

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This page was last edited 23 January 2026.

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