\documentclass[12pt]{article}
\usepackage{amssymb,latexsym}

\def\al{\alpha}
\def\be{\beta}
\def\cof{\mbox{cof}}
\def\de{\delta}
\def\forces{{\;\Vdash}}
\def\ga{\gamma}
\def\la{\langle}
\def\om{\omega}
\def\proof{\par\noindent Proof:\par}
\def\qed{\par\noindent QED\par\bigskip}
\def\ra{\rangle}
\def\su{\subseteq}

\newtheorem{theorem}{Theorem}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{cor}[theorem]{Corollary}

\begin{document}

\begin{lemma}(Bukovsky?, Namba?)
Suppose 
$$p\forces\tau\in\om_3$$ 
then there exists $\ga<\om_3$ and
$q\leq_0p$ such that
$$q\forces\tau<\ga$$
\end{lemma}
\proof
To simplify notation assume the root of $p$ is trivial.
Define $s\in p$ is good iff there exists $\ga<\om_3$ and $q\leq_0p_s$
such that $q\forces\tau<\ga$.  Note that if $s$ is
bad then there are at most $\om_1$ immediate extensions
of $s$ which are good.
So it easy to build
$q\leq_0 p$ such that all $s\in q$ are bad.
But this is a contradiction because then no extension of $q$ 
decides $\tau$.
\qed

Given that $p\forces \tau:\om\to\om_3$, then by an easy fusion
argument there exists $q\leq_0p$ and $\al_s<\om_3$ for $s\in q$ such that
$q_s\forces\tau(|s|)<\al_s$.  Letting
$$\al_{\infty}=\sup\{\al_s:s\in q\}$$ we get
$$q\forces \forall n\;\;\tau(n)<\al_\infty$$
So in the Namba extension the cofinality of $\om_3^M$ is not $\om$.

\begin{theorem}
(Bukovsky) Suppose $M\su N$ are models of ZFC such that
\begin{itemize}
\item $P(\om)^M=P(\om)^N$,
\item $N\models \cof(\om_2^M)=\om$, and
\item $M\models 2^\om=\om_1+2^{\om_1}=\om_2$.
\end{itemize}
Then
$N\models |\om_3^M|\leq \om_1$.
\end{theorem}

\proof
In $M$ choose a sequence 
$\la F_\al:P(\al)\to\om_2 :\al<\om_2\ra$ of  one-to-one maps.
Also choose $\la X_\be\su\om_2:\be<\om_3\ra$ to be distinct.

In $N$ let $\la \de_n:n<\om\ra$ be cofinal in $\om_2^M$.
Working $N$ find embedding:
$$\om_3^M \to (\om_2^M)^\om$$
as follows
$$\be\mapsto\la X_\be\cap\de_n:n<\om\ra\mapsto
\la F_{\de_n}(X_\be\cap\de_n):n<\om\ra$$
But in $N$ the cardinality of $(\om_2^M)^\om$ is $\om_1$.
\qed

\begin{cor}
If $M$ a model of GCH and $G$ is Namba-generic over $M$, 
then in $M[G]$ the cofinality of
$\om_3^M$ is $\om_1$.
\end{cor}

\end{document}