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%\newtheorem*{fact}{Fact}
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\newtheorem*{ques}{Question}
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\newcommand{\ww}{{\omega^\omega}}



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\newcommand{\lh}{\text{lh}}

\newcommand{\ad}{\mathsf{AD}}
\newcommand{\dc}{\mathsf{DC}}
\newcommand{\zf}{\mathsf{ZF}}
\newcommand{\zfc}{\mathsf{ZFC}}

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\newcommand{\cpa}{\overset{\text{c.u.b.}}{\pa}}



\begin{document}


\setbeamercovered{dynamic}



\title{Partition Properties Under AD}



\author{S.\ Jackson}






\institute{(joint project with {\color{red}A.\ Apter} and {\color{red}B.\ L\"{o}we)}\\[10pt]
Department of Mathematics\\
University of North Texas\\
jackson@unt.edu
}






\date{{\color{red} CUNY}\\
October, 2009}







\begin{frame}
\titlepage
\end{frame}



\section{Introduction}





\begin{frame}
We work in the background theory $\zf+\ad+\dc$.
\bigskip



We recall the Erd\H{o}s-Rado partition notation. 
\bigskip



\begin{defn}
$\kappa \pa (\kappa)^\lambda$ iff for every {\color{red} partition} 
$\sP \colon (\kappa)^\lambda \to \{ 0,1\}$ there is a {\color{red} homogeneous}
set $H \subseteq \kappa$ with $|K|=\kappa$. 
\end{defn}
\bigskip


We say $\kappa$ has the {\color{red} strong} partition property if 
$\kappa \pa (\kappa)^\kappa$.
\bigskip


We say $\kappa$ has the {\color{red}weak} partition property if 
$\forall \lambda <\kappa\ \kappa \pa (\kappa)^\lambda$.
\bigskip

In $\zfc$ no $\kappa$ has an infinite exponent partition property.
\end{frame}







\begin{frame}
\frametitle{Polarized Relation}
Suppose $\{ \kappa_i\}_{i <\theta}$ is an increasing, discontinuous sequence of cardinals
and $\{ \lambda_i\}_{i<\theta}$ is a sequence of exponents ($\lambda_i \leq \kappa_i$). 
\bigskip


A {\color{red} block function} $F$ from $\{ \lambda_i\}$ to $\{ \kappa_i\}$
is an increasing function from $\oplus_{i<\theta} \lambda_i$ 
to $\sup_{i<\theta}\kappa_i$ such that $F$ 
restricted to the copy of $\lambda_i$ maps to $\kappa_i- \sup_{j<i}\kappa_j$. 
\bigskip





We say $\{ \kappa_i\} \pa \{\kappa_i\}^{\lambda_i}$ if for all partitions
$\sP$ of the block functions from $\{\lambda_i\}$ to $\{\kappa_i\}$ 
there is a {\color{red}block set} $\{ H_i\}_{i<\theta}$ with 
$H_i \subseteq \kappa_i-\sup_{j<i}\kappa_j$
and $|H_i|=\kappa_i$ which is homogeneous for $\sP$.
\bigskip


If the sequence is finite we write for example 

$$
(\kappa_0,\kappa_1,\kappa_2) \pa (\kappa_0,\kappa_1,\kappa_2)^{\lambda_0,\lambda_1,\lambda_2}
$$

\end{frame}







\begin{frame}

{\color{blue}General Problem}: Determine the exact partition properties for all cardinals
and polarized properties for sequences of cardinals assuming  $\ad$. 
\bigskip


Down low, say below $\aleph_{\omega_1}$, we can use the detailed inductive analysis
of the cardinals. This relies on a detailed theory of the projective ordinals
(essentially  the first few Suslin cardinals)
\bigskip


Up higher we must use general methods relying on the general theory of Suslin cardinals
and pointclasses under $\ad$.
\bigskip


{\color{blue}Conjecture}: Every regular Suslin cardinal has the strong partition
property.



\end{frame}




\begin{frame}
Recall the following fundamental definition of descriptive set theory.
\bigskip



\begin{defn} 
$A \subseteq \ww$ is $\kappa$-Suslin if there is a tree 
$T \subseteq (\omega \times \kappa)^{<\omega}$ with 
$$A=p[T]= \{ x \colon \exists f \in \kappa^\omega\ \forall n\ 
(x \res n, f \res n) \in T\}.$$ Let $S(\kappa)$ be the $\kappa$-Suslin sets.

$\kappa$ is a {\color{red} Suslin cardinal} if $S(\kappa)- \bigcup_{\lambda<\kappa}S(\lambda)
\neq \emptyset$.
\end{defn}
\bigskip


Under $\ad$ the first $\omega$ Suslin cardinals are: 
$$\omega, \quad \omega_1=\de 1 1, \quad \omega_\omega, \quad \omega_{\omega+1}=\de 1 3, 
\quad \omega_{\omega^{\omega^\omega}}, \quad \omega_{\omega^{\omega^\omega}+1}=\de 1 5,
\dots.$$
\bigskip
\end{frame}



\begin{frame}
The corresponding Suslin classes enumerate the first $\omega$ many 
{\color{red} Levy} classes closed under $\exists^{\R}$.
\bigskip

So, $S(\omega)=\bs^1_1$, $S(\de 1 1)=\bs^1_2$, $S((\de 13)^{-})=\bs^1_3$, 
$S(\de 13)=\bs^1_4$, etc. 
\bigskip


In general, $S((\de 1{2n+1})^{-})=\bs^1_{2n+1}$ and 
$S(\de 1{2n+1})=\bs^1_{2n+2}$.
\bigskip


\begin{defn}
$\de 1 n$ is the supremum of the lengths of the $\bd^1_n$ prewellorderings of the reals.
\end{defn}
\bigskip

\begin{fact}
(Kechris) \quad $\cof((\de 1{2n+1})^{-})=\omega$ and \\ 
(Kunen-Martin)\quad  $\de 1{2n+2}=(\de 1{2n+1})^+$.
\end{fact}
\end{frame}



\begin{frame}
\begin{thm}[J]
All the $\de 1{2n+1}$ have the strong partition property. These are the
only cardinals below the projective ordinals with the strong partition property.
\end{thm}
\bigskip


\begin{thm}[J]
Between $\de 1{2n+1}$ and $\de 1{2n+3}$ there are exactly $2^{n+1}-1$
regular cardinals $\kappa$. They all satisfy 
$\kappa \pa (\kappa)^{\de 1{2n+1}}$ but no higher.
\end{thm}
\bigskip




\begin{thm}[Kechris-Kleinberg-Moschovakis-Woodin]
Every regular {\color{red}limit} Suslin cardinal has the strong partition 
property.
\end{thm}
\bigskip

Proof shows many non-Suslin cardinals have the strong partition property.
\end{frame}



\begin{frame}
\begin{thm}
For every regular limit (inaccessible) Suslin cardinal $\kappa$ 
we have 
$$
(\kappa,\kappa^+,\kappa^{++}) \pa (\kappa,\kappa^+,\kappa^{++})^{\kappa, \kappa,\kappa}.$$
\end{thm}
\bigskip



This was part of joint work with A.\ Apter  and B.\ L\"{o}we.
\bigskip



{\color{blue}Conjecture}: For every inaccessible Suslin Cardinal $\kappa$ we have:

$$
(\kappa,\kappa^+,\kappa^{++}) \pa (\kappa,\kappa^+,\kappa^{++})^{\kappa, \kappa^+,\kappa^+}.$$



\begin{rem}
We cannot increase the last exponent to $\kappa^{++}$.
\end{rem}
\bigskip
\end{frame}




\begin{frame}
We consider the possible cofinalities/measurability of $\aleph_1$, $\aleph_2$, $\aleph_3$. 
\bigskip


\begin{itemize}
\item
There are $3$ possibilities of $\aleph_1$ ($\cof=\omega$, $regular$, measurable).
\item
There are $4$ possibilities for $\aleph_2$, and $5$ possibilities for $\aleph_3$.
\item
$13$ of the $60$ total possibilities are ``trivially inconsistent.''
For example, $\aleph_1$ regular, $\cof(\aleph_2)=\aleph_1$, and $\cof(\aleph_3)=\aleph_2$.
\end{itemize}




\begin{thm}
Assuming suitable large cardinals, all of the remaining $47$ patterns are consistent
with $\zf$. 
\end{thm}
\end{frame}










\section{Preliminaries}

\begin{frame}
The following definition is used frequently.
\bigskip



\begin{defn}
We say $f \colon \alpha \to \on$ has {\color{red}uniform cofinality} $\omega$
if there is an $f' \colon \alpha \times \omega \to \on$, inceasing in the second argument, such that 

$$
\forall \beta <\alpha\ f(\beta)=\sup_m f'(\beta,m)$$

We say $f$ is of the {\color{red}correct type} if $f$ is increasing, 
everywhere discontinuous, and of uniform cofinality $\omega$.
\end{defn}
\bigskip



We can also define $f$ having uniform cofinality $\rho$, or more generally 
$f$ having uniform cofinality $g$ for $g \colon \alpha \to \on$. 
\bigskip

For example, $f(\alpha)$ has uniform cofinality $\alpha$.
\end{frame}



\begin{frame}
When proving partition properties from $\ad$ we always use an alternate form of the definition.
\bigskip



\begin{defn}
$\kappa \cpa (\kappa)^\lambda$ if for every partition $\sP$
of the functions $f \in (\kappa)^\lambda$ of the correct type, there is a c.u.b.\
$C \subseteq \kappa$ homogeneous for $\sP$,
\end{defn}



{\color{blue}Fact:}

\begin{equation*}
\begin{split}
& \kappa \cpa (\kappa)^\lambda \Rightarrow  \kappa \pa (\kappa)^\lambda
\\ 
& \kappa \pa (\kappa)^{\omega \cdot\lambda} \Rightarrow  \kappa \cpa (\kappa)^\lambda
\end{split}
\end{equation*}




In particular, for any $\kappa$ the notions of weak or strong 
partition property are unambiguous. 
We officially adopt the c.u.b.\ versions of these properties henceforth.
\end{frame}



\begin{frame}

\begin{thm}[Steel-Woodin]
The Suslin cardinals are closed below their supremum. 
Assuming $\ad^+$, they are closed.
\end{thm}
\bigskip


So, on the c.u.b.\ set of limit Suslin cardinals, $\kappa$ has the strong partition property
iff $\kappa$ is regular. 
\bigskip


Off this set things are more difficult.
\end{frame}


\begin{frame}
All known proofs of partition properties use a general method of {\color{red}Martin}.
\bigskip


Martin's theorem says roughly that to show $\kappa \pa (\kappa)^\lambda$ 
it suffices to code functions $f \colon \lambda \to \kappa$ within 
some pointclass $\bg$ having some reasonable boundedness properties.
\bigskip


For example, Martin  showed $\omega_1 \pa (\omega_1)^{\omega_1}$
using this method and the pointclass $\bg=\bs^1_1$.
\bigskip

To show $\de 13 \pa (\de 13)^{\de 13}$ one uses the pointclass 
$\bg=\bs^1_3$. 
\end{frame}



\section{Inaccessible Suslin Cardinals}


\begin{frame}
Let $\kappa$ be an inaccessible Suslin cardinal less than the supremum of the
Suslin cardinals. 
\bigskip


Since $\kappa$ has the weak partition property, for each regular 
$\rho <\kappa$ there is a $\rho$-cofinal normal measure $\mu_\rho$ on $\kappa$.
\bigskip

In particular, $\mu=\mu_\omega$ is a normal measure on $\kappa$.
\bigskip

\begin{rem}
There are many more normal measures on $\kappa$ than these. 
For every {\em thin} stationary set $S \subseteq \kappa$ there is a normal
measure $\mu_S$ which concentrates on $S$. Such stationary sets are 
well-ordered. The order-type is much  larger than $\kappa$.
\end{rem}

\end{frame}



\begin{frame}

General pointclass arguments show the following.
\bigskip


\begin{fact}
There is a pointclass $\bg$ closed under $\forall^\R$, $\wedge$, $\vee$
and with $\sca(\bg)$. The norms $\varphi_i$ of a $\bg$-scale
on a $\bg$-complete set $P$ map onto $\kappa$. 
\end{fact}
\bigskip


Since $\kappa$ is a limit of Suslin cardinals, $\bd=\bg \cap \bgd$
is closed under real quantification. 
\bigskip


There is a c.u.b.\ $C\subseteq \kappa$ of limit Suslin cardinals
and every $\alpha \in C$ is {\em strongly reliable}. That is, 
if $\beta <\alpha$ then $\sup \{ \varphi_n(x) \colon \varphi_0(x) \leq \beta\}
<\alpha$. 
\bigskip


Let $C_\omega=\{ \alpha \in C \colon \cof(\alpha)=\omega\}$. 

\end{frame}




\begin{frame}
\frametitle{Some local pointclasses}

For $\alpha \in C_\omega$, let $\bs^\alpha_0$ be the pointclass of countable unions
in $S(<\alpha)$, 
\bigskip


Then: $\sca(\bs^\alpha_0)$, $\sca(\bp^\alpha_1)$, $\sca(\bs^\alpha_2)$, etc.
\bigskip


$\bp^\alpha_1$ admits scales with norms onto $\alpha^+$.
\bigskip




$\alpha^+$ is the supremum of the lengths of the $S(\alpha)$ wellfounded relations.
\bigskip



$S(\alpha)=\bs^\alpha_1$, $S(\alpha^+)=\bs^\alpha_2$.
\end{frame}




\begin{frame}

We have the following picture.
\bigskip





\begin{center}
\begin{picture}(250,150)(0,-125)
\put(0,0){\line(1,0){250}}

\put(175,-5){\line(0,1){10}}
\put(173,-12){$\kappa$}


\put(195,-5){\line(0,1){10}}
\put(193,-12){$\kappa^+$}


\put(215,-5){\line(0,1){10}}
\put(213,-12){$\kappa^{++}$}




\put(75,-5){\line(0,1){10}}
\put(85,-5){\line(0,1){10}}
\put(90,-5){\line(0,1){10}}
\put(93,-5){\line(0,1){10}}

\put(95,-5){\line(0,1){10}}
\put(93,-12){$\alpha$}

\put(110,-5){\line(0,1){10}}
\put(108,-12){$\alpha^+$}


\put(125,-5){\line(0,1){10}}
\put(123,-12){$\alpha^{++}$}


\put(150,-12){$\cdots$}

\put(95,0){\fcolorbox{black}{red}{\makebox(8,0.5){}}}



\end{picture}
\end{center}
\vspace{-100pt}

$$
\bs^\alpha_0=\bigcup_{\lambda<\alpha} S(\lambda)
$$

$$
S(\alpha)=\bs^\alpha_1, \qquad S(\alpha^+)=\bs^\alpha_2
$$



$$
\sca(\bs^\alpha_0,) \qquad \sca(\bp^\alpha_1)
$$

\end{frame}


\section{Ultrapower Representation}



\begin{frame}

\begin{lem}

We have the following representations for $\kappa^+$, $\kappa^{++}$.
\bigskip



$$[\alpha \mapsto \alpha^+]_\mu =\kappa^+, \qquad 
[\alpha \mapsto \alpha^{++}]_\mu =\kappa^{++}$$
\end{lem}
\bigskip


Let $R$ be the binary relation corresponding to $\varphi_0$.



$$R(x,y) \leftrightarrow x,y \in P \wedge \varphi_0(x) \leq \varphi_0(y).$$



For $\alpha \in C$ let:



$$
R^\alpha(x,y)  \leftrightarrow x,y \in P \wedge \varphi_0(x) \leq \varphi_0(y)<\alpha.$$
\end{frame}

\begin{frame}
\frametitle{Uniform Sets and Scales}

Using the $R^\alpha$ we uniformly define sets 

$$
A^\alpha \in \bs^\alpha_0, \qquad B^\alpha \in \bs^\alpha_1, \qquad 
Q^\alpha \in \bp^\alpha_1
$$


$A_\alpha(\langle \tau,z \rangle)\leftrightarrow  \exists n\ (\tau(z))_n \in R^\alpha$.
\bigskip


$B^\alpha(\langle \tau,z \rangle) \leftrightarrow  
\exists w \ \forall n\ R^\alpha(\tau(z,w,n))$.
\bigskip

$Q^\alpha(w)\leftrightarrow \forall z\ A^\alpha(\langle w,z\rangle)$.
\end{frame}



\begin{frame}
We can uniformly assign scales and Suslin representations to the $A^\alpha$,
$B^\alpha$, and $Q^\alpha$. 


\begin{itemize}
\item
The scale $\varphi_n$ on $P$ gives a scale $\sigma_n$ on $R$ and by restriction scales on the
$R^\alpha$. 
\item
From the scale $\sigma_n$ on $R$ we easily get a Suslin representation 
$A^\alpha=p[U^\alpha]$ which then gives a $\bs^\alpha_0$-scale on $A^\alpha$. 
\item
Similarly we get a Suslin representation $B^\alpha=p[V^\alpha]$, where 
$V^\alpha$ is a tree on $\omega \times \alpha$. 
\item
From periodicity and the scale on $A^\alpha$ we get a $\bp^\alpha_1$-scale on $Q^\alpha$.
\end{itemize}
\end{frame}









\begin{frame}
We code ordinals in $C_\omega$ as follows.
\bigskip


Let $W=\{ x\colon  \forall n\ (x)_n \in P\}$. 
\bigskip



For $x \in W$ let $\psi(x)=|x|=\sup_n \varphi_n(x)$. 

$W_\alpha=\{ x \in W \colon
\psi(x) \leq \alpha)\}$. 
\bigskip



The scale $\varphi_n$ on $P$ easily give a scale on $W$, let $W=p[T_W]$ be
the Suslin representation. 













\end{frame}




\subsection{Two Trees}




\begin{frame}
We define two trees $T^+$ and $T^{++}$ on $\omega \times \kappa$ to analyze the ultrapowers 
$[\alpha \mapsto \alpha^+]_\mu$ and $[\alpha \mapsto \alpha^{++}]_\mu$.
\bigskip



Will have:


\begin{lem}
For any $f \colon \kappa \to \kappa$ with $f(\alpha)<\alpha^+$ there is a $x \in \ww$ with $T^+_x$
wellfounded and such that $\forall^*_\mu \alpha\ f(\alpha)<|T^+_x \res \alpha|$.
\end{lem}
\medskip


This immediately shows that $[\alpha \mapsto \alpha^+]_\mu \leq \kappa^+$, 
and the lower bound follows easily from the normality of $\mu$.
\end{frame}


\begin{frame}
For the tree $T^{++}$ we will have:



\begin{lem}
For any $f \colon \kappa \to \kappa$ with $f(\alpha)<\alpha^{++}$ there is a $x \in \ww$
with $T^{++}_x$ wellfounded such that 
$\forall^*_\mu \alpha\ (f(\alpha)< [\beta \mapsto |T^{++}_x \res \beta|]_{\mu_\alpha})$.
\end{lem}
\bigskip


The lemma and the normality of $\mu_\alpha$ show that 
$[\alpha \mapsto \alpha^{++}]_\mu \leq \kappa^{++}$.
\medskip


{\color{blue}Note}: We don't immediately get the lower bound.
\medskip



These lemmas will also give a coding of the functions into $\kappa^+$ or $\kappa^{++}$
we use to get the partition relations.
\end{frame}


\subsection{Construction of T+}


\begin{frame}
\frametitle{Construction of $T^+$}


Recall $B^\alpha$ is universal for $\bs^\alpha_1$. From the {\color{red}Kunen-Martin} theorem
and the coding lemma we have that $\alpha^+$ is the supremum of the lengths of the
$\bs^\alpha_1$ wellfounded relations. We code these using $B^\alpha$. 
\bigskip





We define a tree $U^2$ on $\omega^2 \times (\omega^2\times \kappa^2)$ as follows. 
\bigskip

A branch $(x,y,z,w, \vec{\alpha},\vec{\beta})$ through $U^2$ should witness (for $x \in W$):
that $z_0,z_1,\dots$ is a decreasing chain in the $\bs^{\psi(x)}_1$ relation coded
by $y$. This relation is the set of $(c,d)$ such that 

$$
\exists w\ \forall n\ \langle y(\langle c,d \rangle, w \rangle_n =(e,f) \in R^{\psi(x)}
$$


\end{frame}


\begin{frame}

The $\vec{\alpha}$, $\vec{\beta}$ witness this as follows:


\begin{itemize}
\item
The ordinals $\vec{\alpha}$ witness that all the $(e,f)$ are in $R$. 
\item
The $\vec{\beta}$ witness that for each $(e,f)$ there is an $n$ such that 
$(f,x_n) \in R$.
\end{itemize}
\bigskip

{\color{red}Key Point:} For $x \in W_\alpha$ (possibly $\psi(x)<\alpha$)
and $y$ coding a $\bs^{\psi(x)}_1$ wellfounded relation, the tree 
$U^2_{x,y} \res \alpha$ is wellfounded (not just $U^2_{x,y}\res \psi(x)$).
\end{frame}



\begin{frame}
Suppose $f \colon \kappa \to \kappa$ and $f(\alpha)<\alpha^+$ for $\alpha \in C$. 
\bigskip



Consider the game $G_f$:
\bigskip


\begin{center}
\begin{tabular}{rl}
I & $r$ \\
II & $x,y$
\end{tabular}
\end{center}
\bigskip

II wins the run iff 

$$(r \in W) \rightarrow [x \in W \wedge \psi(x)\geq \psi(r) \wedge 
|B^{\psi(x)}_y| >f(|x|)].$$
\end{frame}





\begin{frame}
A boundedness argument shows that II has a winning strategy. 
\bigskip

This suggests the following definition of the tree $T^+$ on 
$(\omega)^2 \times \kappa \times (\omega)^2 \times \kappa$:
\bigskip




$(\sigma, r,  \vec \alpha, x,y, \vec \beta) \in [T^+]$ iff:


\begin{enumerate}
\item
$(r, \vec \alpha) \in [T_W]$.
\item
$\sigma(r)=(x,y)$
\item
$(x,y,\vec \beta) \in [U^2]$.
\end{enumerate}
\bigskip


Then $T^+_\sigma$ is wellfounded and $|T^+_\sigma \res \alpha|>f(\alpha)$ 
for $\mu$ almost all $\alpha$. 
\end{frame}



\subsection{Construction of T++}


\begin{frame}
\frametitle{Construction of $T^{++}$}


We first construct a tree $V^2$ on $(\omega)^2 \times \kappa$
with the following properties:


\begin{enumerate}
\item
$(x,y) \in [V^2]$ iff $x \in W$ and for all $n$, $(y)_n$ codes a $\bs^{|x|}_1$
wellfounded relation $B^{\psi(x)}_{(y)_n}$.
\item
If $x \in W$, $\psi(x) \in C$ then there is a c.u.b.\ $D \subseteq \alpha^+$ 
such that if $\gamma \in D$, $y \in \ww$  and for all $n$\  $(y)_n$
codes a $\bs^{\psi(x)}_1$ wellfounded relation of rank $<\gamma$, then
$V^2_{x,y} \res \gamma$ is illfounded.
\end{enumerate}
\bigskip


\color{red} main point: \color{black}
We can translate the $\bp^{\psi(x)}_1$ statement asserting the wellfoudedness
of the $B^{\psi(x)}_{(y)_n}$ into $\bp^\beta_1$ statements for any $\beta \geq \psi(x)$
(use the $(x)_i$ as in the definition of $U$).


\end{frame}




\begin{frame}
Suppose $x \in W$, $\psi(x) =\alpha\in C$, and $g \colon \alpha^+ \to \alpha^+$. 
\bigskip
Play the game $G_g$:


\begin{center}
\begin{tabular}{rl}
I & $z$ \\
II & $w$
\end{tabular}
\end{center}



II wins the run iff: 

$$
(\forall n\ B^{\alpha}_{(z)_n} \text{ is wellfounded }) \rightarrow 
(B^\alpha_w  \text{ is wellfounded } \wedge |B^\alpha_w| > g( \sup_n 
|B^\alpha_{(y)_n}|) )
$$

\end{frame}






\begin{frame}
By boundedness, II has a winning strategy $\tau$ for any $G_g$. 
\bigskip

Suppose now $f \colon \kappa \to \kappa$ with $f(\alpha)< \alpha^{++}$.
\bigskip









Play the game $G_f$:


\begin{center}
\begin{tabular}{rl}
I & $r$ \\
II & $x,\tau$
\end{tabular}
\end{center}
\bigskip



$r,x$ will be in $W$ and $\tau$ will be strategy  for a game $G_g$ where $[g]_{\mu_\alpha} >f(\alpha)$,
where $\alpha=\psi(x)$.
\end{frame}












\begin{frame}
More precisely, II wins the run iff:


\begin{equation*}
\begin{split}
r \in W & \rightarrow (x \in W \wedge \psi(x) =\alpha \geq \psi(r)) \\ &\wedge \forall y\ 
[ \forall n\ B^\alpha_{(y)_n} \text{ is wellfounded } \rightarrow 
\\ & \qquad B^\alpha_{\tau(y)} \text{ is wellfounded }  \wedge 
|B^\alpha_{\tau(y)} | \geq g( \sup_n |B^\alpha_{(y)_n}|) ]
\end{split}
\end{equation*}


\noindent
for some $g \colon \alpha^+ \to \alpha^+$  with $[g]_{\mu_\alpha} \geq f(\alpha)$.

\end{frame}







\begin{frame}
II has a winning strategy $\sigma$ for any $f$, and this suggests the definition of $T^{++}$:
\bigskip



$(\sigma, r, \vec \alpha, x, \tau, y, z,\vec \beta, \vec \gamma) \in T^{++}$ iff:


\begin{enumerate}
\item
$(r, \vec \alpha) \in [T_W]$.
\item
$\sigma(r)=(x,\tau)$.
\item
$(x,y,\vec \beta)\in [V^2]$
\item
$\tau(y)=z$.
\item
$(x,z,\vec \gamma) \in [U^2]$
\end{enumerate}
\bigskip

The properties of $U^2$ and $V^2$ show that $T^{++}$ has the desired property.
\end{frame}

\section{The Proof}

\subsection{The Countable Case}


\begin{frame}
\frametitle{The countable exponent $\theta$ case.}


Fix a bijection $\pi \colon \omega \cdot \theta \to \omega$.
\bigskip


We code cofinally in $\kappa^+$, $\kappa^{++}$ many ordinals using sections of our trees:
$T^+_x$, $T^{++}_x$.
\bigskip




Suppose $\sP$ is a partition of the block functions from $3 \times \theta$ to 
$(\kappa,\kappa^+,\kappa^{++})$.
\bigskip

Consider the game $G_\sP$:



\begin{center}
\begin{tabular}{rl}
I & $x,y,z$ \\
II & $x',y',z'$
\end{tabular}
\end{center}
\bigskip



\end{frame}







\begin{frame}





(1) If there is an $j<\omega \cdot \theta$ such that $(x)_{\pi(j)} \notin P$ or
$(x')_{\pi(j)} \notin P$, then player~I wins iff for the least such $j$, 
$(x)_{\pi(j)} \in P$.
\medskip



(2) Suppose next that there is an  $\alpha<\kappa$ such that one of the following holds. 

\begin{enumerate}
%\renewcommand{\labelenumi}{(\alph{enumi})}

\item[(a)]  There is a $j< \omega \cdot \theta$ such that either $T^+_{(y)_{\pi(j)}}
\res \alpha$ or $T^+_{(y')_{\pi(j)}}\res \alpha$ is
illfounded.



\item[(b)]  There is a $\beta < \alpha^+$ and a $j < \omega \cdot
\theta$ such that either $T^{++}_{(z)_{\pi(j)}} \res \beta$ 
or  $T^{++}_{(z')_{\pi(j)}} \res \beta $ is
illfounded.
\end{enumerate}



Let $\alpha<\kappa$ be least such that (a) or (b) above holds. 
If (a) holds, let $j$ be least such that (a) holds for $\alpha$ and this $j$. 
In this case, Player I wins provided $T^+_{(y)_{\pi(j)}}$ is wellfounded. 
If (a) does not hold at $\alpha$, but (b) does, let $(\beta,j)$ be lexicographically least
such that (b) holds. Player I wins in this case provided 
 $T^{++}_{(z)_{\pi(j)}} \res \beta$  is wellfounded. 







\end{frame}








\begin{frame}

Assume II has a winning strategy $\tau$.
\bigskip


We define c.u.b.\ sets $C_0 \subseteq \kappa$, $C_1 \subseteq \kappa^+$, and $C_2 \subseteq \kappa^{++}$.
\bigskip


For example, to define $C_2$ we define for $\alpha \in C$, $\beta, \gamma < \alpha^+$
and $j <\omega \cdot \theta$:







\begin{equation*}
\begin{split}
A_{\alpha, \beta,\eta, j}=& \{ (x,y,z) \,\colon \, \forall j\ ( (x)_{\pi(j)}
\in P_0 \wedge \varphi_0((x)_{\pi(j)}) < \alpha) 
\\ & \wedge 
\forall \alpha' < \alpha\ \forall \beta \ <(\alpha')^+\ \forall j\ 
(T^+_{(y)_{\pi(j)}} \res \alpha \text{ and } 
T^{++}_{(z)_{\pi(j)}} \res \beta \text{ are w.f.})
\\ & 
\wedge \forall j \
|T^+_{(y)_{\pi(j)}} \res \alpha| < \beta \wedge 
\forall (\beta',j') \leq_\lex (\beta,j) \ (
|T^{++}_{(z)_{\pi(j)}} \res \beta | \leq \eta ) \}.
\end{split}
\end{equation*}




\end{frame}







\begin{frame}
We have: $A_{\alpha,\beta,\gamma,j} \in \bd^\alpha_1$.
\bigskip





Since $\tau$ is winning for Player~II, for each $(x,y,z) \in
A_{\alpha, \beta, \eta,j}$, if $\tau(x,y,x)= (x',y',z')$ then
$\forall (\beta',j') \leq_\lex (\beta,j)\   
T^{++}_{(z')_{\pi(j')}}\res \beta$ is wellfounded. 
\medskip



By
boundedness, $$\rho_2(\alpha,\beta,\eta,j) := \sup \{ |
T^{++}_{(z')_{\pi(j')}} \res \beta| \,\colon \, (x',y',z') \in
\tau[A_{\alpha,\beta,\eta,j}] \wedge j' \leq j  \} <
\alpha^+.$$
\medskip


Let $C_2^\alpha \subseteq \alpha^+$ be c.u.b.\ closed under $\rho_2$. 
The $C^\alpha_2$ lift to $C_2 \subseteq \kappa^{++}$.
\end{frame}

\subsection{Higher Exponent}

\begin{frame}

We need the following modifications to the countable exponent case.
\bigskip


\begin{itemize}
\item
We code functions $F \colon \kappa \to \kappa$ using the uniform coding lemma. 
\item
We use the Kechris-Woodin theory of generic codes to quantify over codes for ordinals 
less than $\kappa$. 
\item
We modify the payoff of the game as follows:
\end{itemize}
\bigskip


I plays $x$, $y$, $z$, \quad II plays $x'$, $y'$, $z'$
\end{frame}




\begin{frame}
We now consider the least $\alpha<\kappa$ which is bad for one of the players
meaning we check (in this order):


\begin{itemize}
\item
For some $\delta <\alpha$, $y$ or $y'$ is not good at $\alpha$:
for almost all codes $a$ of $\delta$, if $R^\delta_y(a,b)$ then $T^+_b \res \alpha$
is wellfounded. 
\item
For some $\delta<\alpha$ and $\beta<\alpha^+$, $z$ is not good at $(\delta,\alpha,\beta)$:
for almost all codes $a$ of $\delta$, if $R^\delta_z(a,b)$ then 
$T^{++}_z \res \beta$ is wellfounded.
\item
$x$ or $x'$ does not code an ordinal at $\alpha$. 
\end{itemize}


To decide who wins, take least $\alpha$ such that one of these cases holds. 
If first case holds, take least $\delta$. If not, but second case holds
take least $(\beta,\delta)$. 
\end{frame}

\subsection{Another Result}


\begin{frame}
\frametitle{An Extension}


We can also show:

$$
\{ \kappa^{+n}\}_{n \in \omega} \pa (\{ \kappa^{+n} \}_{n \in \omega})^{\kappa}.
$$





















\end{frame}









































\end{document}
