Replacing an ordinal with its cardinality in a partition relation

Question

In The Higher Infinite, Kanamori claims that if $\alpha$ is a cardinal, and $\beta \to (\alpha)^\gamma_\delta$ for some $\beta$, then the least such $\beta$ is a cardinal. I can't seem to think of a proof, as bijections don't have to preserve order types. In particular I'm not sure how to, in general, transfer from a $P:[\#\beta]^\gamma \to \delta$ to a $Q:[\beta]^\gamma \to \delta$.

Answer 1

Assuming that $\gamma$ is finite, the argument is fairly simple:

Suppose $\beta\to(\alpha)^\gamma_\delta$, and fix a bijection $f$ between $|\beta|$ and $\beta$. Consider a coloring $c:[|\beta|]^\gamma\to\delta$. Using $f$ , this gives us a coloring $c':[\beta]^\gamma\to\delta$ (here we used that $\gamma$ is finite). We want to argue that there is a $c$-homogeneous set of type $\alpha$.

The coloring $c'$ admits a homogeneous set $H$ of type $\alpha$. But $\alpha$ is a cardinal. Under the bijection $f$, $H$ corresponds to a $c$-homogeneous subset $H'$ of $|\beta|$ of size $\alpha$, and therefore of order type at least $\alpha$.

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Under choice, we may show that if $\gamma$ is infinite, then there are no interesting relations to consider, so we may as well take $\gamma$ to be finite. However, if we do not assume the axiom of choice, now we may have non-trivial partition relations with infinite exponents, and I expect some restrictions are needed on $\gamma$ for the result to hold.

The problem is that if we do not assume that $\gamma$ is finite, it is no longer automatic that $c$ induces a coloring of $[\beta]^\gamma$. The issue is that $f$ may map some subsets of $|\beta|$ of type $\gamma$ to sets of other order types, and vice versa, some subsets of $\beta$ of type $\gamma$ may not be the image of sets of type $\gamma$.