\documentclass{beamer}
\usepackage{beamerthemesplit}
\usepackage[all]{xy}


\usepackage{amsmath}
\usepackage{amssymb}
\usepackage[all]{xy}
\usepackage{eepic}
\usepackage{txfonts}



\theoremstyle{theorem}
\newtheorem*{thm}{Theorem}
\newtheorem*{lem}{Lemma}


\theoremstyle{definition}
%\newtheorem*{fact}{Fact}
\newtheorem*{defn}{Definition}
\newtheorem*{ques}{Question}
\newtheorem*{cor}{Corollary}
\newtheorem*{ex}{Example}
\newtheorem*{claim}{Claim}











\newcommand{\ww}{{\omega^\omega}}



\newcommand{\res}{\restriction}
\newcommand{\bg}{\boldsymbol{\Gamma}}
\newcommand{\bgd}{\check{\boldsymbol{\Gamma}}}
\newcommand{\bd}{\boldsymbol{\Delta}}
\newcommand{\bs}{\boldsymbol{\Sigma}}
\newcommand{\bp}{\boldsymbol{\Pi}}

\newcommand{\de}[2]{\boldsymbol{\delta}^{#1}_{#2}}


\newcommand{\on}{\text{On}}
\newcommand{\sP}{\mathcal{P}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\R}{\mathbb{R}}



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

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

\newcommand{\cof}{\text{cof}}
\newcommand{\lr}{L(\mathbb{R})}
\newcommand{\sca}{\text{scale}}
\newcommand{\lex}{\text{lex}}

\newcommand{\mys}{\bigskip}
\newcommand{\hod}{\text{HOD}}
\newcommand{\gch}{\text{GCH}}
\newcommand{\ot}{\text{o.t.}}
\newcommand{\dom}{\text{dom}}

\newcommand{\sv}{\mathsf{v}}
\newcommand{\wli}{\mathbf{wlift}}
\newcommand{\sli}{\mathbf{slift}}

\newcommand{\sR}{\mathcal{R}}
\newcommand{\sA}{\mathcal{A}}

\newcommand{\ran}{\text{ran}}
\newcommand{\sbR}{\bar{\sR}}

\newcommand{\cub}{\text{c.u.b.}}
\newcommand{\sep}{\text{sep}}
\newcommand{\bU}{\overline{U}}
\newcommand{\conc}{{}^\smallfrown}



\begin{document}


\setbeamercovered{dynamic}



\title{New Partition Properties under $\ad$}



\author{S.\ Jackson}




\institute{
Department of Mathematics\\
University of North Texas\\
jackson@unt.edu
}






\date{{\color{red} ESI Workshop}\\
June, 2009, Vienna}






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


\section{Introduction}


\begin{frame}

We work in the  theory $\zf+\dc+\ad$. 
\mys



Recall the Erd\H{o}s-Rado Partition notation.
\mys


\begin{defn}
$\kappa \rightarrow (\kappa)^\lambda$ if for every partition $\sP \colon 
(\kappa)^\lambda \to \{0,1\}$ of the increasing functions from $\lambda$ to 
$\kappa$ into two pieces, there is a homogeneous $H \subseteq \kappa$ of
size $\kappa$.
\end{defn}
\mys


We say $\kappa$ has the {\color{red} strong} partition property if $\kappa 
\rightarrow (\kappa)^\kappa$, and say $\kappa$ has the {\color{red} weak}
partition property if $\kappa \rightarrow (\kappa)^\lambda$ for all $\lambda<\kappa$.
\mys


We usually use a reformulation of the partition property which uses c.u.b.\ homogeneous 
sets. 
\mys

To state this, we need the notion of {\color{red} type} of a function. 
\mys
\end{frame}


\subsection{Uniform Cofinality and Types}


\begin{frame}


\begin{defn}
We say $f \colon \alpha \to \on$ has {\color{red} uniform cofinality} $\omega$
if there is an $f' \colon \alpha \times \omega \to \on$ which is increasing 
in the second argument and $f(\beta)=\sup_n f'(\beta,n)$ for all $\beta <\alpha$.
\end{defn}

\begin{defn}
$f \colon \alpha \to \on$ is of the {\color{red} correct type} if 
$f$ is increasing, everywhere discontinuous, and of uniform cofinality $\omega$.
\end{defn}
\mys

We can likewise define uniform cofinality $\omega_1$, $\omega_2$, etc.





\end{frame}



\begin{frame}
\frametitle{General Types}

More generally, for any $g \colon \alpha \to \on$ we can define $f$ having {\color{red}uniform
cofinality $g$}: there is an 
$$f' \colon \{ (\beta,\gamma) \colon \beta < \alpha, \gamma<g(\beta) \}$$
with $f(\beta)= \sup_\gamma f'(\beta,\gamma)$.
\mys

We use frequently the ``almost everywhere'' versions of these notions with respect
to some measure $\mu$ on $\dom(f)$.
\end{frame}

\subsection{Partition Relations}

\begin{frame}

\frametitle{Partition Relations}

\begin{defn}
$\kappa \overset{\cub}{\rightarrow} (\kappa)^\lambda$ 
iff for every partition $\sP$ of the function $f \colon \lambda \to \kappa$ of the correct type, 
there is a c.u.b.\ $C \subseteq \kappa$  which is homogeneous for $\sP$.
\end{defn}
\mys

The ordinary and c.u.b.\ version of the partition relation are essentially equivalent:

\begin{fact}
$$\kappa \overset{\cub}{\rightarrow} (\kappa)^\lambda \Rightarrow 
\kappa \rightarrow (\kappa)^\lambda$$

$$\kappa \rightarrow (\kappa)^{\omega \cdot \lambda} \Rightarrow \kappa \overset{\cub}{\rightarrow}
(\kappa)^\lambda$$
\end{fact}
\end{frame}


\subsection{Suslin Cardinals}

\begin{frame}


\begin{defn}
$A \subseteq \ww$ is {\color{red}$\kappa$-Suslin} if there is a tree $T \subseteq \omega \times \kappa$
such that $A=p[T]= \{ x \colon \exists f \in \kappa^\omega \in (x,f) \in [T]\}$.
Let $S(\kappa)=$ the $\kappa$-Suslin sets.

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


\begin{thm}[Steel-Woodin]
The Suslin cardinals are closed below their supremum.
\end{thm}
\mys


Assuming $V=\lr$, there is a largest Suslin cardinal $\de 21$ (Martin-Steel),
and the Suslin cardinals are c.u.b.\ in $\de 21$. 
\mys
\end{frame}


\begin{frame}
The first $\omega$ many Suslin cardinals are:


\begin{equation*}
\begin{split}
& \lambda_1=\omega, \ \de 11=\omega_1,  \ \lambda_3=\omega_\omega, \ \de 13=\omega_{\omega^\omega+1},
\ \lambda_5=\omega_{\omega^{\omega^\omega}}, \ \de 15= \omega_{\omega^{\omega^\omega}+1},\\ & 
\dots, \lambda_{2n+1}, \ \de 1{2n+1},\dots
\end{split}
\end{equation*}
\mys


Here $\de 1{2n+1}=(\lambda_{2n+1})^+$, and $\lambda_{2n+1}$ is a cardinal of cofinality 
$\omega$ (Kechris).
\mys





Recall $\de 1{2n+2}=(\de 1{2n+1})^+$, and all the $\de 1n$ are regular (measurable).
\end{frame}


\subsection{Polarized Partition Relations}

\begin{frame}




\begin{defn}
Given an increasing, discontinuous sequence of cardinals $\{ \kappa_i\}_{i<\theta}$
and a sequence of ordinals $\{ \lambda_i\}_{i <\theta}$, we say a {\color{red} block function}
from $\vec \lambda$ to $\vec \kappa$ is a $\vec f=\{ f_i\}_{i <\theta}$ where 
$f_i \colon \lambda_i \to \kappa_i- \sup_{j<i} \kappa_j$. A block c.u.b.\ set
is a $\vec C=\{ C_i\}_{i<\theta}$ where $C_i$ is c.u.b. in $\kappa_i-\sup_{j<i} \kappa_j$. 
\end{defn}
\mys

We next define the {\color{red} polarized} partition property.
\mys



\begin{defn}
$\vec \kappa \rightarrow (\vec \kappa)^{\vec \lambda}$  if for every partition 
$\sP$ of the block functions $\vec f \colon \vec \lambda \to \vec \kappa$ into two pieces, 
there is a block c.u.b.\ set $\vec C$ homogeneous for $\sP$. 
\end{defn}
\mys


For sequences of length $3$ we will write  $(\kappa_0,\kappa_1, \kappa_2)
\rightarrow (\kappa_0,\kappa_1,\kappa_2)^{\lambda_0,\lambda_1,\lambda_2}$.
\end{frame}


\section{Main Results}


\begin{frame}


\frametitle{Main Results}


\begin{thm}[Apter, J, L\"{o}we]
Let $\kappa$ be an inaccessible Suslin cardinal. Then 
$(\kappa,\kappa^+,\kappa^{++}) \rightarrow (\kappa,\kappa^+,\kappa^{++})^{\kappa,\kappa,\kappa}$.
\end{thm}
\mys

This extends a result of {\color{red}Kechris} for  the countable exponent case.
\mys



\begin{thm}
For all regular $\kappa$ with $\de 1{2n+1}<\kappa <\de 1{2n+3}$ we have 
$\kappa \rightarrow (\kappa)^{\de 1{2n+1}}$ but 
$\kappa \nrightarrow (\kappa)^{\de 1{2n+2}}$.
\end{thm}
\mys


\begin{cor}
$\aleph_{\omega \cdot 2 +1}$, $\aleph_{\omega^\omega+1}$ are regular cardinals 
without the weak partition property.
\end{cor}
\end{frame}




\begin{frame}
\frametitle{Application}
In [Apter, J, L\"{o}we] we used the first theorem to force over models of $\zf+\ad$ 
to change the cofinalities of $\aleph_1$, $\aleph_2$, $\aleph_3$. 
\mys


\begin{itemize}
\item
There are $3$ possibilities for  $\aleph_1$ Namely, $\cof=\omega$, $r$ (regular, non-measurable), 
or $m$ (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}
\mys







\begin{thm}[Apter, J, L\"{o}we]
Assuming suitable large cardinals, all of the remaining $47$ cases
are consistent with $\zf$.
\end{thm}
\end{frame}



\section{More on Suslin Cardinals}


\begin{frame}




\frametitle{Types of Suslin Cardinals}

By a {\color{red} L\'{e}vy} pointclass we mean a pointclass $\bg$
closed under $\exists^\ww$ or $\forall^\ww$ (or both),
\mys



The Wadge hierarchy of L\'{e}vy pointclasses falls into {\color{red} projective hierarchies} of $4$ types.
\mys

We specialize to the Suslin pointclasses. The limit Suslin cardinals $\kappa$ correspond
to the bases of projective hierarchies. 
\mys


Note that if $\kappa$ is a limit Suslin cardinal then $\Lambda=S(<\kappa)$ 
is a selfdual pointclass closed under quantifiers.
\mys

Also, $o(\Lambda)=\boldsymbol{\delta}(\Lambda)=\kappa$.
\end{frame}



\begin{frame}
If $\cof(\kappa)>\omega$, there is 
a nonselfdual pointclass $\bg_S$ (Steel pointclass) of Wadge degree $\kappa$ 
closed under $\forall^\ww$, $\wedge$,  
with $\sep(\bgd_S)$. Also, $\sca(\bg_S)$.
\mys


\begin{fact}[Steel]
If $\kappa$ is regular, then $\bg_S$ is closed under $\vee$, $\wedge$. 
\end{fact}
\mys


Wren $\kappa$ is regular, we have boundedness of $\bd_S$ sets with respect to 
$\bg_S$-norms on $\bg_S$-complete sets.



\end{frame}



\begin{frame}



\begin{itemize}
\item
{\color{red} Type I} $\cof(\kappa)=\omega$. 
Let $\bs^\kappa_0=\bigcup_\omega (S(<\kappa))$. Then 
$\sca(\bs^\kappa_0)$, $\sca(\bp^\kappa_1)$, $\sca(\bs^\kappa_2), \dots$.
The Suslin cardinals are $\kappa=\lambda^\kappa_1$, $\kappa^+=\de  \kappa 1$, 
$\lambda^\kappa_3$, $\de \kappa 3, \dots$. 
\item
{\color{red} Type II} $\cof(\kappa)>\omega$, $\bg_S$ not closed under $\vee$. 
\item
{\color{red} Type III} $\cof(\kappa)>\omega$, $\bg_S$ closed under $\vee$
but not $\exists^\ww$ $\kappa$ necessarily regular). 
\item{\color{red} Type IV} $\cof(\kappa)>\omega$, $\bg$ closed under $\exists^\ww$,
$\forall^\ww$.
\end{itemize}
\mys


For $\kappa$ an inaccessible Suslin cardinal we are in Type III or Type IV. 
\mys

The Type I hierarchies will play an important role in the proof.
\end{frame}


\begin{frame}
We fix the inaccessible Suslin cardinal $\kappa$. Let $C \subseteq \kappa$
be the c.u.b.\ set of limit Suslin cardinals.Let $C_\omega \subseteq C$
be the points of cofinality $\omega$. 
\mys

$\kappa$ has the strong partition property by Kechris-Kleinberg-Moschovakis-Woodin.
\mys

Let $\mu$ be the $\omega$-cofinal, normal measure on $\kappa$. 
\mys


For $\alpha \in C_\omega$, let $\mu_\alpha$ be the $\omega$-cofinal, normal measure on 
$\alpha^+$. We have 
$j_{\mu_\alpha}(\alpha^+)=\alpha^{++}$. 
\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{The Proof}
\subsection{Plan of the Proof}


\begin{frame}



\frametitle{Plan of the proof}






\begin{enumerate}
\item
We show $[\alpha \mapsto \alpha^+]_\mu=\kappa^+$.


\item
We show $\delta \doteq [\alpha \mapsto \alpha^{++} ]_\mu \leq \kappa^{++}$.


\item
We show for all $\theta < \omega_1$ that $(\kappa,\kappa^+,\delta)\rightarrow
(\kappa,\kappa^+,\delta)^\theta$.

\qquad It follows that $\delta$ is regular, so $\delta =\kappa^{++}$.




\item
Finally, we show $(\kappa,\kappa^+,\delta)\rightarrow
(\kappa,\kappa^+,\delta)^\kappa$
\end{enumerate}


\end{frame}

\subsection{The Trees $T^+$, $T^{++}$}

\begin{frame}


\frametitle{The Trees $T^+$ and $T^{++}$}
We define two trees $T^+$ and $T^{++}$ on $\omega \times \kappa$. 
\mys


\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}
\mys



\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
\mys

The first lemma shows $[\alpha \mapsto \alpha^+]_\mu\leq \kappa^+$, and it follows easily
that $[\alpha \mapsto \alpha^+]_\mu = \kappa^+$.
\mys

The second lemma shows that $\delta \doteq [\alpha \mapsto \alpha^{++}]_\mu \leq \kappa^{++}$.

\end{frame}



\subsection{Construction of the Trees}

\begin{frame}



\frametitle{Construction of $T^+$}
Fix a $\bg$-complete set $P$ and a $\bg$-scale $\{ \varphi_n \}_{n \in \omega}$ on $P$. 
we use $\varphi=\varphi_0$ to code ordinals $< \kappa$. 
\mys



Say $\alpha$ is  {\color{red} strongly reliable} if for all $\beta < \alpha$:


$$
\sup\{ \varphi_n(x) \colon x \in P \wedge \varphi_0(x) \leq \beta\} <\alpha
$$


The set of strongly reliable ordinals  is c.u.b.\ in $\kappa$. Assume $C \subseteq$
strongly reliables.
\mys




Let 

$(x,y) \in R \leftrightarrow (x,y \in P \wedge \varphi(x) < \varphi(y)\}$.
\mys

and  

$(x,y) \in R^\alpha  \leftrightarrow (x,y \in P \wedge \varphi(x) < \varphi(y)<\alpha\}$.

\end{frame}



\begin{frame}
$R^\alpha  \in \bs^\alpha_0- \bigcup_{\lambda < \alpha} S(\lambda)$,
and we uniformly have a $\bs^\alpha_0$ scale on $R^\alpha$ (essentially by restricting the
scale $\vec \varphi$ to ordinals below $\alpha$). 
\mys




Starting from this, we uniformly get $\bs^\alpha_1$ universal sets $B^\alpha$
and $\bp^\alpha_1$ sets $Q^\alpha$ and $\bp^\alpha_1$ scales $\vec \psi^\alpha$ on $Q^\alpha$. 
\mys



\begin{defn}
Let $W=\{ x \colon \forall n\ (x)_n \in P\}$. $x \in W$ will code the ordinal 
$|x|=\sup_n \varphi_0((x)_n)$. 
\end{defn} 
\mys



The scale on $P$ easily gives a scale on $W$. Let 
$T_W$ be the corresponding tree.

\end{frame}

\begin{frame}
We first construct a tree $U$ on $\omega \times \omega \times \kappa$
with the following properties:

\begin{enumerate}
\item
If $x \in W$ and $|x| =\alpha \in C$, then $U_{x,y}$ is wellfounded iff 
the $\bs^\alpha_1$ relation coded by $y$ is wellfounded. 

\item
For $x,y$ as above, $|U_{x,y} \res \alpha| \geq |B^{\alpha}_y|$, the 
$\bs^\alpha_1$ relation coded by $y$. 
\end{enumerate}
\bigskip



\color{red} Key Point: \color{black} For $x,y$ as above, the entire tree 
$U_{x,y}$ is wellfounded (not just $U_{x,y} \res \alpha$).
\bigskip


\color{red} idea: \color{black}
$U$ is constructed as in the proof of the Kunen-Martin theorem, but we use the components of the real 
$x$ to verify the appropriate reals are in $B^{|x|}_y$.

\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 

$$(r \in W) \rightarrow (x \in W \wedge B^{|x|}_y  \text{ is wellfounded
} \wedge |B^{|x|}_y| >f(|x|).$$
\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
$(x,y,\vec \beta) \in [U]$.
\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 $V$ on $(\omega)^2 \times \kappa$
with the following properties:


\begin{enumerate}
\item
$(x,y) \in [V]$ iff $x \in W$ and for all $n$, $(y)_n$ codes a $\bs^{|x|}_1$
wellfounded relation $B^{|x|}_{(y)_n}$.
\item
If $x \in W$, $|x| \in C$ 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 $\bs^{|x|}_1$ wellfounded relation of rank $<\gamma$, then
$V_{x,y} \res \gamma$ is illfounded.
\end{enumerate}
\bigskip


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


\end{frame}




\begin{frame}
Suppose $x \in W$, $|x| =\alpha\in C$, 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 $\alpha=|x|$.
\end{frame}












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


\begin{equation*}
\begin{split}
r \in W & \rightarrow (x \in W \wedge |x| =\alpha \geq |y|) \\ &\wedge \forall z\ 
[ \forall n\ B^\alpha_{(z)_n} \text{ is wellfounded } \rightarrow 
\\ & \qquad B^\alpha_{\tau(z)} \text{ is wellfounded }  \wedge 
|B^\alpha_{\tau(z)} | \geq g( \sup_n |B^\alpha_{(z)_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
$(x,y,\vec \beta)\in [V]$
\item
$\tau(y)=z$.
\item
$(x,a,\vec \gamma) \in [U]$
\end{enumerate}
\bigskip

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


\subsection{Proof for 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 $(x)_{\pi(j)} \notin P_0$ or
$(x')_{\pi(j)} \notin P_0$, then player~I wins iff for the least such $j$, 
$(x)_{\pi(j)} \in P_0$.
\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,\gamma, j}=& \{ (x,y,z) \,;\, \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 wellfounded})
\\ & 
\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, $$\rho_2(\alpha,\beta,\eta,j) := \sup \{ |
T^{++}_{(z')_{\pi(j')}} \res \beta| \,;\, (x',y',z') \in
\tau[A_{\alpha,\beta,\eta,j}] \wedge j' \leq j  \} <
\alpha^+.$$
\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{General Exponent  case}

\begin{frame}



\frametitle{Exponent $\kappa$}

We use generic codes ({\color{red}Kechris-Woodin}) and the uniform coding lemma.
\mys


Let $U(Q, z,x,y)$ be universal for the syntactic class $\bs_1(Q)$, where $Q$ is a binary predicate symbol. 
\mys

Can take 

$$
U(Q,z,x,y) \leftrightarrow \exists w\ (S(z, \langle x,y,w \rangle) \wedge 
\forall n\ W(((w)_n)_0, ((w)_n)_1) ).$$

where $S$ is universal $\bs^1_1$. 

\mys



Let $R'_\alpha$ code $\{ (x,y) \colon \varphi(x)<\varphi(y) \leq \alpha\}$ and $ \leq $ version.
\mys


We write $U_z(R'_\alpha)(x,y)$ for $U(R'_\alpha,z,x,y)$.
\mys
\end{frame}


\begin{frame}


{\color{red} Uniform Coding Lemma} says that if $A \subseteq \ww \times \ww$ with $\dom(A)=P$,
then there is a $z \in \ww$ such that for all $\alpha < \kappa$, 
$U_z(R'_\alpha)$ is a choice subrelation for $A \res P_{\leq \alpha}$. 
\mys



Let $\bU_z(R'_\alpha)$ be a uniformization of $U_z(R'_\alpha)$ 
(using scale $\{ \varphi_n\}$.
\mys



For $\alpha<\kappa$, let $\alpha'$ be the next reliable (w.r.t.\ $\{\varphi_n\}$).
\mys


Let $G \colon \kappa^\omega \to \ww$ be a {\color{red}generic coding function}. So for any $s \in \kappa^\omega$,
$x=G(\alpha \conc s) \in P$, $\varphi(x) \leq \alpha$, and if $s$ enumerates an honest set
then $|x|=\alpha$. 
\end{frame}


\begin{frame}

Given reals $x$, $y$, $z$ we say:

\begin{enumerate}
\item
$x$ {\color{red}codes a function} at $\alpha <\kappa$ if $|a|=\alpha$ implies 
$\exists b \in P\ \bU_x(R'_\alpha)(a,b)$. Also, if $|a|=|a'|$ then 
$|b|=|b'|$.
\item
$y$ is {\color{red}good} at $\delta < \alpha \in C_\omega$ if $\forall^* a \in P_\delta\
\exists b\ \bU_y(R'_\delta)(a,b)$ and $T^+_b \res \alpha$ is wellfounded
(good at $\alpha$ if good at all $\delta <\alpha$).
\item
$z$ is {\color{red}good} at $\delta <\alpha \in C_\omega$, $\beta<\alpha^+$ if 
$\forall^* a \in P_\delta\ \exists b\ \bU_z(R'_\delta)(a,b)$ and 
$T^{++}_z \res \beta$ is wellfounded (good at $\alpha$ if good at all 
$\delta <\alpha$, $\beta <\alpha^+$).
\end{enumerate}
\mys


$f_x(\alpha)=$ the unique value of $|b|$.
\mys

$g_y(\delta,\alpha)=$ least $\gamma <\alpha^+$ such that $\forall^* a \in P_\delta, \ 
|T^+_b \res \alpha|\leq \gamma$.
\mys


$h_z(\delta,\alpha,\beta)=$ least $\gamma <\alpha^+$ such that 
$\forall^* a \in P_\delta, \ |T^{++}_b \res \beta| \leq \gamma$.
\end{frame}




\begin{frame}
I plays $x, y, z$, II plays $x', y', z'$. 
\mys

Let $\alpha$ be least such that one of the following holds.
\mys


\begin{enumerate}
\item
For some $\delta <\alpha$, $y$ or $y'$ is not good at $(\delta,\alpha)$.
\item
For some $\delta <\alpha$, $\beta<\alpha^+$, $z$ or $z'$ is not good
at $(\delta,\alpha,\beta)$. 
\item
$x$ or $x'$ does not code a function at $\alpha$. 
\end{enumerate}
\mys


If (1) holds, then I wins iff for the least such $\delta$, $y$ is good at $(\delta,\alpha)$.
Suppose (1) does not hold, but (2) holds. Then I wins iff for the lexicographically least 
pair $(\beta,\delta)$ we have $z$ is good at $\delta, \alpha, \beta$. 
If (1) and (2) don't hold, but (3) holds, then I wins iff $x$ codes an ordinal at $\alpha$.
\end{frame}


\begin{frame}
Otherwise, we have functions $f_x$, $f_{x'}$, $g_y$, $g_{y'}$, $h_z$, $h_{z'}$.
\mys

\begin{itemize}
\item
$f_x$, $f_{x'}$ together produce $F \colon \kappa \to \kappa$. 

\item
$g_y$, $g_{y'}$ together produce $G \colon \kappa \to \kappa^+$.


\item
$h_z$, $h_{z'}$ together produce $H \colon \kappa \to \kappa^{++}$.
\end{itemize}
\mys


II wins the run in this case iff $\sP(F,G,H)=1$.
\mys


From a winning strategy $\tau$ for II, say, we define the homogeneous sets 
$C_0 \subseteq \kappa$, $C_1 \subseteq \kappa^+$ and $C_1 \subseteq \kappa^{++}$.









\end{frame}

































\end{document}
