# Determinacy from $\omega_1\rightarrow(\omega_1)^{\omega_1}$

Assuming the Axiom of Determinacy (abbreviated AD), Martin showed how to derive a rather strong partition on $\omega_1$, namely that $\omega_1\rightarrow(\omega_1)^{\omega_1}$. In "Infinitary Combinatorics and the Axiom of Determinateness", Kleinberg goes through Martin's proof that $\omega_1\rightarrow(\omega_1)^{\omega_1}$ under AD, and from there he goes on to prove much about the large cardinal structure of ZF+AD below $\aleph_\omega$. In fact, he does this directly from the partition relation. In particular, Kleinberg shows that $\aleph_1$ and $\aleph_2$ are measurable, for each $n>2, \aleph_n$ has cofinality $\aleph_2$ and is Jonsson, and that $\aleph_\omega$ is Rowbottom.

We see that the existence of such a partition relation is inconsistent with choice. In fact, the following is a theorem of Erdos and Rado:

For all infinite $\tau$, there is no $\kappa$ such that $\kappa\rightarrow(\tau)^\tau$

However, I am curious about the following:

How much is known about the amount of determinacy one can derive from ZF + DC + $\omega_1\rightarrow(\omega_1)^{\omega_1}$?

In particular, any references would be greatly appreciated.

Edit:

I elaborated on the nature of the question in a comment to an answer. For sake of visibility, I'm pasting the relevant portion here:

To elaborate on the question, it's interesting that one can derive so much about the large cardinal structure of ZF + AD directly from the strong partition relation on $\omega_1$. As far as I know, "$\omega_1$ is measurable" and "$\omega_2$ is measurable" alone allow us a fragment of determinacy. I was curious as to what is known beyond what can be derived from these measures, as $\omega_1\rightarrow(\omega_1)^{\omega_1}$ gives us these measures.

There is some sort of equivalence between partition properties and determinacy. The question is treated in Kechris, Kleinberg, Moschovakis, Woodin, Determinacy, partition properties, nonsingular measures in the Cabal reprints, Volume I. More specifically the weak partition property on $\omega_1$ implies analytic determinacy (which is provable in ZF). Concerning the strong partition property I am not really sure, I think it implies $\Pi^1_1$-determinacy (remember this is equivalent to $0^{\sharp}$ and originally the strong partition on $\omega_1$ was proven using properties of indiscernibles). The KKMW article has the details
• ZFC should be ZF in line 4. Analytic determinacy is $\mathbf \Pi^1_1$-determinacy, so I do not understand the sentence that begins "Concerning..." Plus, the partition property implies measurability, so it cannot be proved "using properties of indiscernibles", I am not sure what you mean here. Aug 1 '13 at 5:34
• I am aware of the proofs. The point is: You claim that the weak partition property implies blah. Then you say that you think that a stronger statement implies blah. Now, indeed the weak partition property implies analytic determinacy because it implies that $\omega_1$ is measurable. Since it implies that $\omega_1$ is measurable, it cannot be proved just using properties of indiscernibles, on consistency strength grounds alone. (I'm pointing this out because the problem of the exact consistency strength of the strong partition property is interesting, and seems delicate.) Aug 1 '13 at 5:56
• (In case it is not clear: $\Sigma^1_1$-determinacy is equivalent to $\Pi^1_1$-determinacy, and it is strictly weaker than $\mathbf\Pi^1_1$-determinacy.) Aug 1 '13 at 5:57
• I think Andres has cleared up most of the confusion here, but maybe the following will still help. Analytic determinacy (equivalent, as Andres said, to co-analytic determinacy) is equivalent to the existence of sharps of reals. So in this sense, it is a matter of indiscernibles. The existence of a measurable carinal is, of course, a strictly stronger statement than existence of sharps of reals. And lightface $\Sigma^1_1$ (or $\Pi^1_1$) determinacy is equivalent to the existence of $0^{\#}$, which is strictly weaker than the existence of sharps of all reals. Aug 1 '13 at 16:40
• Regarding the answer itself, the reference does help a fair bit. My concerns are slightly different from those addressed in that particular paper. Specifically, it's interesting that one can derive so much about the large cardinal structure of ZF + AD directly from the strong partition relation on $\omega_1$. As far as I know, "$\omega_1$ is measurable" and "$\omega_2$ is measurable" alone allow us a fragment of determinacy. I was curious as to what is known beyond what can be derived from these measures, as $\omega_1\rightarrow(\omega_1)^{\omega_1}$ gives us these measures. Aug 1 '13 at 18:32