Can there be an infinite $\kappa$ with $\kappa\to(\omega)^\omega_2$ in $\mathsf{ZF}$?

Question

It is a standard result in $\mathsf{ZFC}$ that $\kappa\not\rightarrow(\omega)^\omega_2$ for infinite $\kappa$, but the proof I've seen requires well ordering $[\kappa]^\omega$ so it doesn't work in $\mathsf{ZF}$.

Can there be an infinite $\kappa$ for which $\kappa\to(\omega)^\omega_2$ holds if we drop the axiom of choice? I'm being vague in what infinite means here because I'm interested in results using any of the notions of infinite available in $\mathsf{ZF}$.

Answer 1

Yes, it is consistent to have such cardinals. In fact, it is consistent relative to an inaccessible cardinal that $\omega\to(\omega)^\omega_2$. This is a famous result of Mathias, in

MR0491197 (58 #10462). Mathias, A. R. D. Happy families. Ann. Math. Logic 12 (1977), no. 1, 59–111.

(It is still open whether the inaccessible cardinal is required.)

The result holds in Solovay's model where all sets of reals are Lebesgue measurable. It also holds under the assumption of $\mathsf{AD}^+$, in particular, in all known models of $\mathsf{AD}$.

$\mathsf{AD}^+$ is a technical strengthening of the axiom of determinacy $\mathsf{AD}$, and it is open whether both theories are actually the same. All our techniques to obtain models of determinacy give us models of $\mathsf{AD}^+$. It is open whether $\mathsf{AD}$ suffices.

Actually, $\mathsf{AD}$ gives us many additional examples of cardinals $\kappa$ with the required partition property, and much more. For instance, determinacy implies that $\omega_1\to(\omega_1)^{\omega_1}_2$. This is the strong partition property. A good deal of infinitary combinatorics under $\mathsf{AD}$ is about establishing such partition properties at various cardinals. The consistency strength of such an uncountable (well-orderable) cardinal is higher than for $\omega\to(\omega)^\omega_2$, as the assumption readily gives us that $\omega_1$ and $\omega_2$ are measurable cardinals which, in turn, gives us inner models of $\mathsf{ZFC}$ with two measurable cardinals (and more).

I suggest Jech's set theory book or Kanamori's The higher infinite to learn more about the subject. There is a very nice short book by Kleinberg devoted solely to consequences of these partition properties,

MR0479903 (58 #109). Kleinberg, Eugene M. Infinitary combinatorics and the axiom of determinateness. Lecture Notes in Mathematics, Vol. 612. Springer-Verlag, Berlin-New York, 1977. iii+150 pp. ISBN: 3-540-08440-1.