%S. Shelah, L. Soukup:
%The existence of large  ${\omega}_1$-homogeneous
%but not ${\omega}$-homogeneous
%permutation groups  is consistent with ZFC + GCH,
%written in LaTeX2e
\documentstyle[12pt]{article}
\author{S. Shelah
\thanks{The first author was supported by the United States Israel Binational
Science Foundation, Publication number 389}
\\Institute of Mathematics\\Hebrew University, Jerusalem \and L. Soukup
\thanks{The second author was supported by the
Hungarian National Foundation for Scientific Research grant no. l805}
\\Mathematical Institute of the\\Hungarian Academy of Sciences}
\title{The existence of large  ${\omega}_1$-homogeneous
but not ${\omega}$-homogeneous
permutation groups  is consistent with ZFC + GCH}
\begin{document}

\maketitle
%\input{trandef}
\def\<{\left\langle}
\def\>{\right\rangle}
\def\cc{\gamma}
\def\dd{\delta}
\def\n{\nu}
\def\m{\mu}
\def\br#1;#2;{\bigl[ {#1} \bigr]^ {#2} }
\def\bc#1;#2;{\bigl( {#1} \bigr)^ {#2} }
\def\k{\kappa}
\def\Dom{\mbox{dom}}
\def\Ran{\mbox{ran}}
\def\supp{\mbox{supp}}
\def\ooseq#1;#2;{\< {#1}_{#2}:{#2}<\oo\>}
\def\ooset#1;#2;{\{ {#1}_{#2}:{#2}<\oo\}}
\def\seq#1;#2;#3;{\< {#1}_{#2}:{#2}<#3\>}
\def\set#1;#2;#3;{\{ {#1}_{#2}:{#2}<#3\}}
\def\oseq#1;#2;{\< {#1}_{#2}:{#2}<\O\>}
\def\oset#1;#2;{\{ {#1}_{#2}:{#2}<\O\}}
\def\oosequ#1;#2;{\< {#1}^{#2}:{#2}<\oo\>}
\def\oosetu#1;#2;{\{ {#1}^{#2}:{#2}<\oo\}}
\def\sequ#1;#2;#3;{\< {#1}^{#2}:{#2}<#3\>}
\def\setu#1;#2;#3;{\{ {#1}^{#2}:{#2}<#3\}}
\def\osequ#1;#2;{\< {#1}^{#2}:{#2}<\O\>}
\def\osetu#1;#2;{\{ {#1}^{#2}:{#2}<\O\}}
\def\fsq#1;#2;{\<#1_0,\dots,#1_{#2}\>}
\def\fsqv#1;#2;{\vec{#1}=\<#1_0,\dots,#1_{#2}\>}
\def\con#1;{\operatorname{Con}(\text{#1})}
%\def\rest{{\tsize |}\raise 5pt \hbox{$@!@!\!\sssize\backslash$}}
\def\rest{\lceil}

\def\is{\cong}
\def\bij{@>1-1>onto>}
\def\abs#1{\|#1\|}
\def\force{\raisebox{1.5pt}{\mbox{$\scriptscriptstyle\|$}}
\mbox{$\!\mbox{---}$}}
\def\qst{Q^{*}}

\def\inc{\not\|}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{Definition}[theorem]{Definition}
\newtheorem{observation}[theorem]{Observation}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{sublemma}{Sublemma}[theorem]


\def\ujbekezdes{\par}
\newenvironment{definition}
{\begin{Definition}\rm}{\end{Definition}}
\newenvironment{demo}[1]
{{\ujbekezdes}\noindent{\bf #1:}\ \rm}{{\ujbekezdes}}
\newenvironment{case}[2]
{{\ujbekezdes}\noindent{\bf Case\ #1:}{\em #2}\ujbekezdes \rm}{{\ujbekezdes}}

\newcommand{\proof}{\begin{demo}{Proof}}
\newcommand{\Proof}[1]{\begin{demo}{Proof of #1}}
\newcommand{\eproof}{\ \rule{2mm}{2mm}\end{demo}\vspace{1mm}}

\newcommand{\alabel}{
          \renewcommand{\labelenumi}{{\rm (\alph{enumi})}}
          \renewcommand{\theenumi}{{\rm (\alph{enumi})}}
                    }
\newcommand{\Alabel}{
          \renewcommand{\labelenumi}{{\rm (\Alph{enumi})}}
          \renewcommand{\theenumi}{{\rm (\Alph{enumi})}}
                    }
\newcommand{\rlabel}{
          \renewcommand{\labelenumi}{{\rm (\roman{enumi})}}
          \renewcommand{\theenumi}{{\rm (\roman{enumi})}}
                    }
\newcommand{\Rlabel}{
         \renewcommand{\labelenumi}{{\rm(\Roman{enumi})}}
         \renewcommand{\theenumi}{{\rm(\Roman{enumi})}}
                    }


\newcommand{\litem}{\renewcommand{\labelitemi}{{\bf --}}}


\newdimen\elso
\newdimen\masodik
\newdimen\harmadik
\def\tagg#1#2$${
\settowidth{\elso}{\hbox{$\displaystyle #2$}}
\settowidth{\masodik}{\hbox{$\displaystyle (#1)$}}
\harmadik \textwidth
\advance\harmadik by -\elso
\advance\harmadik by -\masodik
\advance\harmadik by -\masodik
\divide\harmadik by 2
\hbox{$\displaystyle (#1)$}
\makebox[\harmadik]{}
\hbox{$\displaystyle #2$}
\makebox[\harmadik]{}
\makebox[\masodik]{}
$$}


\def\multline{\begin{array}{c}}
\def\endmultline{\end{array}}



\newcommand{\limo}{\mbox{\rm {Lim}}}
\newcommand{\leo}{<_{\mbox{\rm{On}}}}
\newcommand{\oo}{{{\omega}_1}}
\newcommand{\acal}{{\cal A}}
\newcommand{\bcal}{{\cal B}}
\newcommand{\ccal}{{\cal C}}
\newcommand{\dcal}{{\cal D}}
\newcommand{\fcal}{{\cal F}}
\newcommand{\gcal}{{\cal G}}
\newcommand{\hcal}{{\cal H}}
\newcommand{\ical}{{\cal I}}
\newcommand{\jcal}{{\cal J}}
\newcommand{\kcal}{{\cal K}}
\newcommand{\mcal}{{\cal M}}
\newcommand{\ncal}{{\cal N}}
\newcommand{\pcal}{{\cal P}}
\newcommand{\rcal}{{\cal R}}
\newcommand{\qcal}{{\cal Q}}
\newcommand{\scal}{{\cal S}}
\newcommand{\tcal}{{\cal T}}
\newcommand{\wcal}{{\cal W}}
\newcommand{\xcal}{{\cal X}}
\newcommand{\atil}{\widetilde A}
\newcommand{\btil}{\widetilde B}
\newcommand{\htil}{\widetilde \chi}
\newcommand{\util}{\widetilde U}
\newcommand{\ztil}{\widetilde z}
\newcommand{\abar}{\bar a}
\newcommand{\hhbar}{\bar h}
\newcommand{\bbar}{\bar b}
\newcommand{\ahat}{\hat a}
\newcommand{\bhat}{\hat b}
\newcommand{\chat}{\hat c}
\newcommand{\dhat}{\hat d}
\newcommand{\dplus}{{\diamondsuit^{^{\makebox[-0.1cm]{\tiny +}
\makebox[0.1cm]{}}}}}
\newcommand{\fini}{{<{\omega}}}
\newcommand{\setm}{\setminus}
\newcommand{\empt}{\emptyset}
\newcommand{\subs}{\subset}
\newcommand{\bijp}{\mbox{\rm {Bij}${}_p$}}
\newcommand{\isop}{\mbox{\rm{Iso}${}_p$}}
\newcommand{\fin}{\mbox{\rm{Fin}}}
\newcommand{\fino}{\fin({\omega},2)}
\newcommand{\dom}{\mbox{\rm{dom}}}
\newcommand{\ran}{\mbox{\rm{ran}}}
\newcommand{\gwf}{g^{W,f}}
\newcommand{\Sub}{\mbox{\rm{Sub}}}
\newcommand{\Cl}{\mbox{{\rm Cl}}}
\newcommand{\cont}{(2^{\omega})}
\newcommand{\II}{\mbox{\rm{I}}}
\newcommand{\Gr}{ $G=\langle V,E\rangle$ }
\def\rel#1;{\mbox{rel}_G(#1)}
\def\eqnum#1;#2;{\#(#1/\equiv_{#2})}
\def\typ{\mbox{type${}_G$}}
\def\oot{{{\omega}_2}}
\newcommand{\rank}{\mbox{rank}}
\newcommand{\compp}{\|_{{}_P}}
\def\perm#1;{\mbox{Perm}(#1)}
\def\pperm#1;#2;{\mbox{Perm}^{#2}(#1)}
\newcommand{\permo}{\perm\oo;}
\newcommand{\poo}{{2^{\oo}}}
\newcommand{\poop}{(2^{\oo})^{\scriptscriptstyle +}}
\newcommand{\permpo}{\mbox{Perm}_p(\oo)}
\newcommand{\rvec}{\vec{r}}
\newcommand{\rests}{\rest^*}
\newcommand{\abc}{\<\atil,\btil,C\>}
\newcommand{\abcm}{\<\atil\rests{\mu},\btil\rests{\mu},C\cap{\mu}\>}
\newcommand{\abcn}{\<\atil\rests{\nu},\btil\rests{\nu},C\cap{\nu}\>}
\newcommand{\abci}{\<\atil_i,\btil_i,C_i\>}
\newcommand{\abcj}{\<\atil_j,\btil_j,C_j\>}
\newcommand{\abcnj}{\<\atil^{\nu}_j,\btil^{\nu}_j,C^{\nu}_j\>}
\newcommand{\abcni}{\<\atil^{\nu}_i,\btil^{\nu}_i,C^{\nu}_i\>}
\newcommand{\abcmi}{\<\atil^{\mu}_i,\btil^{\mu}_i,C^{\mu}_i\>}
\newcommand{\abcnjm}{\<\atil^{\nu}_j\rests{\mu},\btil^{\nu}_j\rests{\mu},
C^{\nu}_j\cap {\mu}\>}
\newcommand{\abcel}{\<\atil^{\eta}_l,\btil^{\eta}_l,C^{\eta}_l\>}
\newcommand{\abcelm}{\<\atil^{\eta}_l\rests{\mu},\btil^{\eta}_l\rests{\mu},
C^{\eta}_l\cap {\mu}\>}

\def\rem#1{\mbox{Rem}(N_#1) }
\newcommand{\bijpoo}{\bijp(\oo,\oo)}
\newcommand{\nr}[1]{N_{#1}[\rvec_{#1}]}
\newcommand{\wolog}{Without loss of generality we can assume that   }
\newcommand{\detp}{\mbox{Det}(P_{\oo})}
\newcommand{\uund}[1]{\mbox{und}(#1)}
\newcommand{\und}[1]{\mbox{Und}(#1)}
\newcommand{\undp}[1]{\mbox{Und}^+(#1)}
\newcommand{\undm}[1]{\mbox{Und}^-(#1)}
\newcommand{\HH}{\mbox{H}}
\newcommand{\vppoo}{V^{P_{{\poo}}}}
\newcommand{\vpm}{V^{P_{\mu}}}
\newcommand{\vpn}{V^{P_{\nu}}}
\newcommand{\nj}{\<\nu,j\>}
\newcommand{\mi}{\<\mu,i\>}
\newcommand{\omi}{\omega\mu+i}
\newcommand{\onj}{\omega\nu+j}
\newcommand{\mpip}{\<\mu',i'\>}
\newcommand{\dcd}{\mbox{\rm dcd}}
\def\pand#1;#2;#3;{#1\land_{#2}#3}
\newcommand{\nem}{\neg}


\begin{abstract}
Denote   by $\mbox{Perm}({\lambda})$ the group
of all permutations of a cardinal ${\lambda}$.
A subgroup $G$ of $\mbox{Perm}({\lambda})$
is called {\it ${\kappa}$-homogeneous} iff
for all $X,Y\in [{\lambda}]^{\kappa}$
there is a $g\in G$ with $g''X=Y$. We show that if either
(i) $\dplus$ holds
and  we add ${\omega}_1$ Cohen reals to the ground model,
or (ii) we add $2^{{\omega}_1}$ Cohen reals to the ground model,
then in the generic extension
for each ${\lambda}\ge{\omega}_2$ there is an
${\omega}_1$-homogeneous subgroup
 of $\mbox{Perm}({\lambda})$ which is
 not ${\omega}$-homogeneous. 
\end{abstract}
%\input{tran1}
\section{Introduction}

Denote by $\perm{\lambda};$ the group of all permutations of
a cardinal ${\lambda}$.
The subgroups of $\perm {\lambda};$ are  called
{\em permutation groups on ${\lambda}$}. We say that
a permutation group  $G$ on ${\lambda}$ is  {\em ${\kappa}$-homogeneous}
iff for all  $X,Y\in\br{\lambda};{\kappa};$ with
$|{\lambda}\setm X|=|{\lambda}\setm Y|={\lambda}$ there is
a $g\in G$ with $g''X=Y$.
Given cardinals ${\lambda}$,  ${\kappa}$  and ${\mu}$ write
$\HH({\lambda},{\kappa},{\mu})$ to mean that every ${\kappa}$-homogeneous
permutation group on ${\lambda}$ is ${\mu}$-homogeneous, as well.
P. M. Neumann has raised the problem if
${\lambda}>{\kappa}>{\mu}$ implies $\HH({\lambda},{\kappa},{\mu})$.
He  proved \cite{N} that  $\HH({\lambda},{\kappa},{\mu})$ holds
for $\lambda>{\kappa}\ge{\omega}$ and ${\mu}<{\omega}$.
P. Nyikos \cite{Ny} and independently S. Shelah and
S. Thomas \cite{ST} showed that $\nem\HH(2^{\omega},{{\omega}_1},{\omega})$
is consistent with Martin's Axiom.
Recently  A. Hajnal \cite{H} proved that
$\raisebox{4pt}{\mbox{${\fbox{{}}}_{\,{\omega}_{1}}$}}$ implies
$\nem\HH({\omega}_2,{{\omega}_1},{\omega})$.
The aim of this paper is to construct models of ZFC in which 
$\nem\HH({\lambda},{{\omega}_1},{\omega})$ for each ${\lambda}\ge{\omega}_2$.

We will use the standard notation of  set theory, see \cite{J}.
For sets $A$ and $B$ let us denote by $\fin(A,B)$ 
 the poset whose underlying set consists of all functions mapping
a finite subset of $A$ into $B$ and whose ordering is  the inclusion.


The $\dplus$ principle  asserts that {\em 
there is a sequence 
$\bigl\langle
 S_{\alpha}:{\alpha}<{{\omega}_1}\bigr\rangle$ of countable sets
such that for each $X\subset {{\omega}_1}$ we have a closed unbounded
$C\subset{{\omega}_1}$ with $X\cap{\nu}\in S_{\nu}$ and 
$C\cap\nu\in S_{\nu}$ for each $\nu\in C$.}



 
\begin{theorem}
\label{th:oot}
Assume that either  $(i)$  $\dplus$ holds and 
$P=\fin(\oo,2)$ , or $(ii)$  $P=\fin(\poo,2)$.
Then
\begin{displaymath}
V^P\models
\mbox{``$\nem\HH({\lambda},{{\omega}_1},{\omega})$ for each 
${\lambda}\ge{{\omega}_1}$.''}
\end{displaymath}
\end{theorem}

The proof of this theorem is based on the following observation.
To formulate it we need some definitions. Let $G$ be a permutation
group on $\lambda$. Given $X,Y\subs \lambda$ we say that $X$ is
{\em $(G,Y)$-large} iff for each $n\in\omega$ and $g_0,\dots,g_{n-1}\in G$
we have $|X\setm\bigcup\limits_{i<n}g''_i Y|=|X|$. $G$ is called
{\em $\kappa$-inhomogeneous} iff there are $X,Y\in\br \lambda;\kappa;$ such
that $X$ is $(G,Y)$-large.
\begin{observation} Assume that
$V_0\subs V_1$ are ZFC models and ${\lambda}\ge{\omega}_2$  is
a cardinal in $V_1$. If
\begin{enumerate}\alabel
\item $V_0\models$``$|X|=\oo$'' iff $V_1\models$``$|X|=\oo$''
for each $X\in V_0$. 
\item $\forall X\in\br {\lambda};{{\omega}_1};\cap V_1\ $
$\exists Y\in\br {\lambda};{{\omega}_1};\cap V_0\ $
$X\subs Y$,
\item $V_1\models$
``There is an $\oo$-homogeneous, but $\omega$-inhomogeneous
 \\
\makebox[2cm]{} permutation group $G$ on $\oo$ containing
$\pperm {{\omega}_1}; V_0;$
.''
\end{enumerate}
then $V_1\models$$\nem\HH({\lambda},{{\omega}_1},{\omega})$.
\end{observation}

\Proof{the observation}
We will work in $V_1$.
For each $g\in G$ define
$g^*\in\perm{\lambda};$ by taking $g^*\rest\oo=g$ and 
$g^*(\alpha)=\alpha$ for $\alpha\in{\lambda}\setm\oo$.
Let
$G^*$ be the subgroup of $\perm {\lambda};$ generated by the set
$\{g^*:g\in G\}\cup \pperm \lambda;V_0;$.
We claim that  $G^*$ witnesses $\nem\HH(\lambda,\oo,\omega)$.
First we show that $G^*$ is not $\omega$-homogeneous. Pick
$X,Y\in\br\oo;\omega;$ such that $X$ is $(G,Y)$-large.
%Assume on the contrary
%that $X=g''Y$ for some $g\in G$. Write
%$g=h^*_0\circ p_0\circ h^*_1\circ p_1\circ\dots\circ h^*_{n-1}\circ p_{n-1}$
%where $p_i\in\pperm \lambda;V_0;$ and $h_i\in G$.
We show that  $X$ is  $(G^*,Y)$-large.
Assume on the contrary that $X\subs\bigcup\limits_{i<n}g_i''Y$,
where $g_i\in G^*$. Then
there must be an $i<n$ such that $X\cap g_i''Y$ is $(G,Y)$-large.
So we can assume that $n=1$, i.e, $X\subs g''Y$ for some $g\in G^*$.
Write
$g=h_0\circ(g_0)^*\circ h_1\circ\dots\circ (g_{m-1})^*\circ h_m$,
where $h_i\in\pperm\lambda;V_0;$ and $g_i\in G$. We can
assume that $X$ and $Y$ were  chosen such that $m$  is minimal.
Define the function $d:Y\to m$ by the equation
$$d(y)=\max\{j:(h_j\circ(g_j)^*\circ\dots\circ h_m)(y)\in\oo\}.$$
Take $Y_j=d^{-1}\{j\}$ for  $j\le m$. For $y\in Y_j$ we have
$(h_j\circ(g_j)^*\circ\dots\circ h_m)(y)=(h_j\circ
h_{j+1}\circ\dots\circ h_m)(y)$. Since $\pperm\oo;V_0;\subs G$,
for each  $j\le m$
there is an $f_j\in G$ such that $f_j(y)=(h_j\circ
h_{j+1}\circ\dots\circ h_m)(y)$ for  each $y\in Y_j$. Putting
$W_j=f_j''Y_j$ and $Z_j=g_{j-1}''W_j=(g_{j-1}^*)''W_j$ this means that
$$X\subs \bigcup\limits_{j\le m}
(h_0\circ(g_0)^*\dots\circ h_{j-1})''Z_j.$$
But $Z_j\subs (g_{j-1}\circ f_j)''Y$, $g_{j-1}\circ f_j\in G$,  
so there is an  $j\le m$ such that 
taking 
$$X^*=
X\cap (h_0\circ(g_0)^*\dots\circ h_{j-1})''Z_j$$ we have that  $X^*$ is
$(G,Z_j)$-large.  But  
$X^*\subs (h_0\circ(g_0)^*\dots\circ h_{j-1})''Z_j$ 
 and 
$j-1<m$, which contradicts  the
minimality of $m$.
So $G^*$ is  ${\omega}$-inhomogeneous.

If $X,Y\in\br{\lambda};{{\omega}_1};$ first pick 
$X_0,Y_0\in\br{\lambda};{{\omega}_1};\cap V_0$ with $X\subs X_0$ and
$Y\subs Y_0$ such that $|X_0\setminus X|=|Y_0\setminus Y|={{\omega}_1}$.
Fix $f,h\in\pperm{\lambda}; V_0;$ with $f''X_0={{\omega}_1}$
and $h''Y_0={{\omega}_1}$.
Since $G$ is ${{\omega}_1}$-homogeneous, we have a $g\in G$ with
$g''(f''X)=h''Y$. Then $(h^{-1}\circ g\circ f)''X=Y$ and
$h^{-1}\circ g\circ f\in G^*$.
\eproof

By this observation  the following theorem yields theorem
\ref{th:oot}.
\begin{theorem} \label{th:aux} If either $(i)$
$\dplus$ holds and $P=\fin(\oo,2)$, or  $(ii)$ $P=\fin(\poo,2)$
then
$V^P\models$
{\em ``there is an $\oo$-homogeneous, but $\omega$-inhomogeneous
permutation group $G$ on $\oo$ with
$G\supset\pperm \oo;V;$''.}
 \end{theorem}


\section{Proof of theorem 1.3}
%\input{sketch}
Since the proof of theorem \ref{th:aux} is quite long and technical,
first we sketch the main ideas.

In both cases we will define an iterated forcing system
with finite support
$$\<P_{\nu}:-1\le{\nu}\le\kappa,Q_{\nu}:-1\le{\nu}<\kappa\>$$
and an increasing sequence of permutation groups
$\<G_{\nu}:\nu<\kappa\>$,
$G_{\nu}\prec\pperm \omega_1;{V^{P_{\nu}}};$, simultaneously,
where $\kappa=\oo$ in case $(i)$ and
${\kappa}=\poo$ in case $(ii)$.

Take $G_0=\pperm\oo;V;$ and $P_0=Q_{-1}=\fin({\omega},2)$.
Denote by $r$ the Cohen real  given by the $V$-generic
filter over $P_0$. By standard density arguments it is easy to see that
${\omega}$ is $(G_0,r)$-large.
We  try to carry out the inductive construction of the sequence
$\seq G;{\nu};{\kappa};$ in such a way that
\begin{enumerate}\alabel
\item For each ${\nu}<{\kappa}$ we pick
$X_{\nu}$, $Y_{\nu}\in(\br\oo;\oo;)^{V^{P_{\nu}}}$, then we construct
a permutation
$g_{\nu}\in V^{P_{\nu}+1}$ with $g_{\nu}''X_{\nu}=Y_{\nu}$ and
take   $G_{{\nu}+1}$ as the subgroup of
$\pperm \oo;V^{P_{{\nu}+1}};$ generated by
$G_{\nu}\cup\{g_{\nu}\}$,
\item ${\omega}$ is $(G_{\nu},r)$-large,
\end{enumerate}
and we hope that the sequence $\seq G;{\nu};{\kappa};$
gives us a required permutation group.
In case $(ii)$ we
use a bookkeeping function to ensure
that every pair $X,Y\in(\br{\oo};{\oo};)^
{V^{P_{2^{\omega_1}}}}$ with
$|{\oo}\setm X|=|{\oo}\setm Y|={\oo}$  will be chosen
as $X_\nu$, $Y_\nu$ in some step.
Then $G=\bigcup_{{\nu}<{\poo}}G_{\nu}$ will
be $\oo$-homogeneous.
So the question is whether we can preserve (b) during the induction.
In case $(i)$ we can pick only $\oo$-many pairs $\<X,Y\>$,
so we can't expect that
G$'=\bigcup_{{\nu}<{\oo}}G_{\nu}$ is $\oo$-homogeneous.
But in
this case we will use $\dplus$ principle to pick
the sets $X_{\nu}$, $Y_{\nu}$ for ${\nu}<\oo$ in such a way that
if
we  consider the elements of $G'$
as the family of countable approximations of our required group,
and  we  take
$$
G=\{g\in\pperm\oo;V^{P_{\oo}};:\forall {\nu}<\oo\,
\exists g'\in G'\, g\rest{\nu}=g'\rest {\nu}\},
$$
then $G$ will be $\oo$-homogeneous.
Obviously,  ${\omega}$ is $(G,r)$-large
provided ${\omega}$ is $(G',r)$-large.
%On the other hand, we will use the $\dplus$ principle
%to pick the sets $X_{\nu}$ and $Y_{\nu}$ such that $G^*$
%is $\oo$-homogeneous.

So the question is whether one can preserve $(b)$.
Unfortunately we can't prove a preservation theorem for
$(G,r)$-largeness of $\omega$, but we will introduce the notion
of  goodness of a pair $(G,r)$ which can be preserved during
a suitable iteration and it will clear  from
the definition that the goodness of
$(G,r)$ implies that ${\omega}$ is $(G,r)$-large.

After this introduction let us see the details.





%\Proof{theorem \ref{th:aux}}
%
For $g\in\perm\oo;$ and $r\subs{\oo}$ we will write, by an abuse of
notation,
$g(r)$ for $g''r$.
Given  sets $X$ and $Y$ let us denote by
$\bijp(X,Y)$  the set of all bijections between subsets of $X$
and $Y$.

If  $a_0$, $a_1$, $b_0$ and $b_1$ are subsets of $\oo$ with
$a_0\cap a_1=b_0\cap b_1=\empt$ take
 \begin{displaymath}
Q_{a_0,a_1,b_0,b_1}=\{g_0\cup g_1:g_i\in\bijp(a_i,b_i)
\mbox{ and }|g_i|<{\omega}
\mbox{ for }i\in 2\}
\end{displaymath}
and $\qcal_{a_0,a_1,b_0,b_1}=\<Q_{a_0,a_1,b_0,b_1},\supseteq\>$.


\begin{definition}
Let $\fcal \subs\bijp(\oo,\oo)$ and  $r\subs {\omega}$
be given.
\begin{enumerate}
\item A sequence $\xcal=\seq x;{\nu};{\mu};\subs\br\oo;<{\omega};$
is called {$(\fcal,r)$-large } iff
for each $n\in{\omega}$ and  $ h_0,\ldots,h_{n-1}\in \fcal$
there exists a ${\nu}<{\mu}$ such that
\begin{displaymath}
x_{\nu}\cap\bigcup\{h_i(r):i<n\}=\empt.
\end{displaymath}
\item  We say that the  pair  $(\fcal,r)$ is  {\em good}
iff every uncountable sequence  $\ooseq y;{\nu};$ of pairwise disjoint, finite
subsets of $\oo$ has a countable initial segment
$\seq y;{\nu};{\mu};$  which is
$(\fcal,r)$-large.
%\item A sequence $\xcal=\seq x;{\xi};{\nu};\subs\br\oo;<{\omega};$
%is called {$(\fcal,r)$-huge } iff
%for each ${\mu}<{\nu}$
%the sequence $\<x_{\xi}:{\mu}\le{\xi}<{\nu}\>$ is $(\fcal,r)$-large.
\end{enumerate}
\end{definition}


\begin{case}{(i)}{\ $\dplus$ holds and $P=\fin(\oo,2)$.}
We  will define an iterated forcing system
$$\<P_{\nu}:-1\le{\nu}\le\oo,Q_{\nu}:-1\le{\nu}<\oo\>$$ with finite support,
such that
$P_0=Q_{-1}=\fin(\omega,2)$ and for each
${\nu}<\oo$  either $Q_{\nu}=\fin(\omega,2)$ or
$V^{P_{\nu}}\models$ ``{\it
$Q_{\nu}=\qcal_{A^{\nu}_0,A^{\nu}_1,B^{\nu}_0,B^{\nu}_1}$ 
for some infinite, countable  subsets 
$A^{\nu}_0$, $A^{\nu}_1$, $B^{\nu}_0$, $B^{\nu}_1$ of $\oo$''}.
Since $V^{P_\nu}\models$``{\it the completions of $Q_\nu$ and
$\fin(\omega,2)$ are isomorphic}'', we have 
$V^P=V^{P_{\oo}}$ , and so we may construct our desired
permutation group in  $V^{P_{\oo}}$.
For $\nu\le\oo$ let $\gcal_{\nu}$ be the $P_{\nu}$-generic filter.
Take $g=\cup\gcal_0:\omega\to 2$ and $r=g^{-1}\{1\}$.

Fix  a large enough regular cardinal  ${\kappa}$ and 
let $\ooseq N;{\nu};$ be a sequence of countable, elementary
submodels of $\hcal_{\kappa}=\<H_{\kappa},\in,\prec\>$,
where  $H_{\kappa}$ consists of the sets whose transitive closure
has cardinality $<{\kappa}$ and $\prec$ is a well-ordering of $H_{\kappa}$,
with the property that
$\seq N;{\mu};{\nu};\in N_{\nu}$ for each ${\nu}<\oo$.
%Denote by $\preceq$ the lexicographical  ordering of $\oo\times\omega$.
Fix a $\dplus$-sequence $\scal=\ooseq S;{\alpha};\in N_0$.

If $\nu\le\mu<\omega_1$ and $\gcal$ is a $P_\nu$-generic filter,
then take
$$
N_{\mu}[\gcal]=\{K_{\gcal}(x):x\in N_\mu\}.
$$



%For ${\nu}<\oo$ and $n<{\omega}$
%let ${\xi}_{{\nu},n}={\omega}+{\omega}^2{\nu}
%\stackrel{\raisebox{-2pt}{.}}{-}1)
%+{\omega}n$
%and  $P_{\nu}=\fin({\xi}_{{\nu},0},2)$.

A $P_{\oo}$-name $\htil$ of the  characteristic function of a subset
of $\oo$ is called a {\em canonical ${\nu}$-name} iff
${\nu}\le\oo$ and there exist maximal antichains
$D^{\htil}_{\xi}\subs P_{\oo}$ and functions
$h^{\htil}_{\xi}:D^{\htil}_{\xi}\to 2$ for ${\xi}\in{\nu}$
such that
\begin{displaymath}
\htil=\{\<p,\widehat{\<{\xi},i\>}\>:i=h^{\htil}_{\xi}(p),
p\in D^{\htil}_{\xi},{\xi}\in{\nu}\}.
\end{displaymath}
$\htil$ is called  {\em nice ${\nu}$-name} iff $\htil$ is a canonical
${\nu}$-name and
$\bigcup\limits_{{\xi}<{\nu}}D^{\htil}_{\xi}\subs P_{\omega\nu}$.
We will usually identify sets and their characteristic functions.
To simplify our notation we will write $p\force$``${\rho}\in\htil$''
for $p\force$``$\htil({\rho})=1$''.


If $\atil$ is a nice $\nu$-name and $\gcal$ is a
$P_{\omega\nu}$-generic filter, then write
$$\atil[]gcal]=\{\xi\in\nu:\exists p\in \gcal\,p\force
\xi\in\atil\}.$$


Given a $P_{\oo}$-name $\atil$  and ${\nu}<\oo$ take
\begin{displaymath}
\atil\rests{\nu}=\{\<p,x\>\in\atil:p\in P_{\omega\nu}\}.
\end{displaymath}

A triple $\abc$ is called {\em ${\nu}$-remarkable} iff
$C\subs{\nu}$ is a club, $\atil\rests{\mu}$  and $\btil\rests{\mu}$ are
nice ${\mu}$-names and $\abcm\in N_{\mu}$ for each ${\mu}\in C\cup \{{\nu}\}$.

Take
\begin{displaymath}
\rem{\nu}=\left\{\abc:\abc\mbox{ is ${\nu}$-remarkable}\right\}.
\end{displaymath}

%Let $\gcal$ be $P$-generic over $V$.
%Take $g=\cup\gcal$, $r=g\rest {\xi}_{0,0}$ and
%$r_{{\nu},n}=g\rest [{\xi}_{{\nu},n},{\xi}_{{\nu},n+1})$
%for ${\nu}<\oo$ and $n<{\omega}$.
%Write $\gcal_{\omega\nu}=\<r,r_{{\mu},m}:{\mu}<{\nu}, m<{\omega}\>$
%and
%$\rvec_{\nu,n}=\<r,r_{{\mu},m}:{\mu}<{\nu}\mbox{ and } m<{\omega}\mbox{ or }
%{\mu}={\nu}\mbox{ and }m<n\>$.

First we construct the family $\fcal'$ of ``countable approximations''
of elements  of our required permutation group.


By induction on ${\nu}$ we will define the posets
$Q_{\omega\nu+i}$ for $i<\omega$, functions
$f_{\abc}$ for $\abc\in \rem{\nu}$
and families $\fcal_{\nu}\subs\bijpoo$ in $V^{P_{\oo}}$ such that
\begin{enumerate}
\alabel
\item $\fcal_0=\bijpoo\cap V$,
\item $\seq \fcal;{\mu};{\nu};\in N_{\nu}[{\gcal_{\omega\nu}}]$,
\item $\fcal_{\nu}=\bigcup\limits_{{\mu}<{\nu}}\fcal_{\mu}\cup
\left\{f_{\abc}:\abc\in\rem{\nu}\right\}$,
\item $\dom f_{\abc}=\ran f_{\abc}={\gamma}_{\abc}$ for some
${\gamma}_{\abc}\in C\cup\{{\nu}\}$,
\item if $\abc\in \rem{\nu}$, $\mu\in C$, then
$$f_{\abcm}=f_{\abc}\rest \gamma_{\abcm}.$$  
\end{enumerate}

Assume that
$\seq \fcal;{\mu};{\nu};$ is constructed.
From now on we will work in $V[{\gcal_{\omega\nu}}]$. Put
$\fcal_{<{\nu}}=\bigcup\limits_{{\mu}<{\nu}}\fcal_{{\mu}}$.
For  a triple $\abc\in\rem{\nu}$
take \begin{displaymath}
f^*_{\abc}=\bigcup\limits_{{\mu}\in C}f_{\abcm}
\end{displaymath}
and ${\gamma}^*_{\abc}=\sup\limits_{{\mu}\in C}{\gamma}_{\abcm}$.
Clearly $f^*_{\abc}\in N_{\nu}[{\gcal_{\omega\nu}}]$.

Let $\fcal_{{\nu}}^*=\left\{f^*_{\abc}:\abc\in\rem{\nu}\right\}$.
 $N_{\nu}[{\gcal_{\omega\nu}}]\in N_{{\nu}+1}[\gcal_{\omega\nu}]$
implies that
 $\fcal_{\nu}^*\in N_{{\nu}+1}[\gcal_{\omega\nu}]$ and
$\fcal_{\nu}^*$ is countable in $N_{{\nu}+1}[\gcal_{\omega\nu}]$.
So we can fix an enumeration $\fcal_{\nu}^*=\{f^*_{\abcnj}:j<{\omega}\}$
in $N_{{\nu}+1}[\gcal_{\omega\nu}]$.
To simplify our notation we will write
$f^*_{{\nu},j}=f^*_{\abcnj}$.

We  define the posets $Q_{\omega\nu+j}$ and the functions
$f_{\abcnj}\supset f^*_{\abcnj}$ by induction on
$j$.
Assume that we have done it for $i<j$.
Take
\begin{displaymath}
\fcal_{{\nu}}^j=\fcal_{<{\n}}\cup\fcal_{{\nu}}^*\cup\{f_{\abcni}:i<j\}.
\end{displaymath}


Put $I_{\nu,j}=[{\gamma}^*_{\abcnj},{\nu})$. Take $\xi=\omega\nu+j$.
Let $A^{\xi}_0=I_{{\nu},j}\cap \atil^\nu_j[\gcal_{\omega\nu}]$, 
$A^{\xi}_1=I_{{\nu},j}\setm \atil^\nu_j[\gcal_{\omega\nu}]$,
$B^{\xi}_0=I_{{\nu},j}\cap \btil^\nu_j[\gcal_{\omega\nu}]$,
$B^{\xi}_1=I_{{\nu},j}\setm \btil^\nu_j[\gcal_{\omega\nu}]$
and $\qcal^\xi=\qcal_{A^{\xi}_0,A^{\xi}_1,B^{\xi}_0,B^{\xi}_1}$.

If the sets $A^{\xi}_0, A^{\xi}_1, B^{\xi}_0$ and $ B^{\xi}_1$ 
are all $(\fcal_{{\nu}}^j,r)$-large,
then put
$Q_{\xi}=\qcal^\xi$. 
Let $\gcal^\xi$ be the $Q_\xi$-generic filter
over $V^{P_\xi}$. Take $f'_{\abcnj}=\cup\gcal^\xi$ and let
$f_{\abcnj}=f^*_{\abcnj}\cup f'_{\abcnj}$ and so
$\gamma_{\abcnj}=\nu$.

Otherwise, if at least one of 
the sets $A^{\xi}_0, A^{\xi}_1, B^{\xi}_0$ and $ B^{\xi}_1$ is not 
$(\fcal_{{\nu}}^j,r)$-large,
put   $Q_{\omega\nu+j}=\fin(\omega,2)$, $f'_{\abcnj}=\empt$ and $f_{\abcnj}=f^*_{\abcnj}$.

The inductive construction is done.

Take $\fcal=\bigcup\limits_{{\nu}\in\oo}\fcal_{{\nu}}$.
Let $\fcal'$ be the closure of $\fcal$ under composition
and inverse image.


\begin{lemma}
\label{lm:rfgood}
$(\fcal',r)$ is a good pair.
\end{lemma}

Since its  proof  is quite long we postpone it for a while.



Having the family $\fcal'$ of countable approximations
we are ready to construct $G$.


A pair $\<\atil,\btil\>$ is called {\em nice} iff both $\atil$
and $\btil$ are nice $\oo$-names of uncountable, co-uncountable
subset of $\oo$.
Using the
$\dplus$ principle,  for each  nice pair $\<\atil,\btil\>$
choose a club set $C=C_{\atil,\btil}\subs\oo$ such that
$\atil\rests {\nu}$ and $\btil\rests {\nu}$ are
nice ${\nu}$-names and $\abcn\in N_{{\nu}}$ for all ${\nu}\in C$,
and put
\begin{displaymath}
f_{\atil,\btil}=\bigcup\limits_{{\nu}\in C}f_{\abcn}.
\end{displaymath}

Let $G$ be the subgroup of $\perm{\oo};$ generated by the set
\begin{displaymath}
\left\{f_{\atil,\btil}:\mbox{$\<\atil,\btil\>$ is a nice pair
with $\dom f_{\atil,\btil}=\oo$} \right\}.
\end{displaymath}
We claim that $G$ satisfies the
requirements of the theorem.
%Let us remark that we will show that $(G,r)$
%is good without using the $\dplus$ principle.



Since cofinal many  countable initial segments of the elements of $G$
are in $\fcal'$,  lemma \ref{lm:rfgood} implies that
$(G,r)$ is a good pair. Applying this fact for $\oo$ as an uncountable
sequence of pairwise disjoint finite subsets of $\oo$ we obtain that
there is a countable ordinal ${\alpha}$ which is $(G,r)$-large.
% But there is a
%$g_{\alpha}\in\bijpoo\cap V=\fcal_0\subs G$ with
%$g''_{\alpha}({\omega})={\alpha}$, so $\omega$ must be
%$(G,r)$-large.
This means that $G$ is $\omega$-inhomogeneous.

Next we show that  $G$ is ${{\omega}_1}$-homogeneous.
%Up to this point we have not used $\dplus$ principle.
Pick a  nice pair $(\atil,\btil)$. Let $C=C_{\atil,\btil}$ and
consider  the function
$f=f_{\atil,\btil}=\bigcup\limits_{{\nu}\in C}f_{\abcn}$.
If $\dom(f)=\oo$ then $f\in G$ and $f''A=B$, so we are done. Assume that
$\dom(f)={\mu}<\oo$. Then ${\mu}\in C$. But the pair $(\fcal',r)$
is good, so there is a ${\nu}>{\mu}$, ${\nu}\in C$, such that
the sets $({\nu}\setm {\mu})\cap A$, $({\nu}\setm {\mu})\setm A$,
$({\nu}\setm {\mu})\cap B$ and  $({\nu}\setm {\mu})\setm B$
are all $(\fcal',r)$-large.
%So in the ${\nu}^{\mbox{\tiny th}}$ step
%of the construction
%we extended $f^{\abcm}$. So $\dom(f)\supset{\nu}$.
So we get $\dom(f'_{\abcn})=\nu\setm\mu$. Thus $\dom(f)\supset{\nu}$.
Contradiction, $G$ is $\oo$-homogeneous.

%Let us remark that only the proof of $\oo$-homogeneity
%of $G$ required the $\dplus$ principle.

Finally we present the postponed proof of lemma \ref{lm:rfgood}:
\Proof{lemma \ref{lm:rfgood}}
To start with we introduce some
notion.

%If $p\in P$, ${\nu}<\oo$ and $j\in\omega$ take
%$p[{\nu},j]=p\rest[{\xi}_{{\nu},j},{\xi}_{{\nu},j+1})$.
A condition $p\in P_{\oo}$ is called {\em determined} iff
for each $\xi<\oo$ if $p(\xi)\ne 1_{Q_\xi}$, then
$
p\rest {\xi}\force$
``${Q_{\xi}=\fin(\omega,2)}$''  or
$p\rest {\xi}\force$
``${Q_{\xi}=\qcal^\xi}$''
and  there is  a function  $F_{p,\xi}$ in the ground model
%and ${\gamma}^p_{\abcnj}\in\oo$
such that
$p(\xi)=
\widehat{F}_{p,\xi}$.
Put  $$\detp=\{p\in P_{\oo}: p\mbox{ is determined}\}.$$
The proof of the following sublemma
is straightforward:
\begin{sublemma}
\label{lm:det}
$\detp$ is dense in $P_{\oo}$.
\end{sublemma}

Consider a condition $p\in\detp$. For each
$\abcnj\in\rem{\nu}$ take
\begin{displaymath}
\begin{array}{rll}
f^{[p]}_{{\nu},j}&=\bigcup\left\{p(\omega{\mu}+i)\right.:&
p\rest{(\omega\mu+i)}\force
\mbox{``${Q_{\omi}=\qcal^{\omi}}\makebox[2pt]{{}}$''},
{\mu}\in C^{\nu}_j\cup\{\nu\}\mbox{ and }
\\&&\left.\makebox[1.5cm]{{}}\abcmi=\abcnjm\right\}.
\end{array}
\end{displaymath}

The finite function $f^{[p]}_{{\nu},j}$ gathers all the
information contained in $p$ about $f_{\abcnj}$.

An {\em $\fcal$-term $t$} is a sequence
$\<h_0,h_1,\ldots,h_{n-1}\>$ where
 $h_i=f_i$ or $h_i=f_i^{-1}$ for some $f_i\in\fcal$.
We also use $t$ to denote the function
$h_0\circ\cdots\circ h_{n-1}$.
We say that $t'$ is a {\em subterm of $t$} if
$t'=\<h_{i_0},\ldots,h_{i_l}\>$, where
$0\le i_0<\ldots<i_l\le n-1$.


Given an  $\fcal$-term $t=\<h_0,h_1,\ldots,h_{n-1}\>$ and  a condition
$p\in\detp$
take
$t^{[p]}=h'_0\circ\cdots\circ h'_{n-1}$
where
\begin{displaymath}
h'_i=\left\{
\begin{array}{ll}
h_i&\mbox{if }h_i\in\fcal_0\ (=\bijpoo\cap V),\\
f^{[p]}_{{\nu},j}&\mbox{if $h_i=f_{\abcnj}$ for some ${\abcnj}\in\rem{\nu}$},\\
(f^{[p]}_{{\nu},j})^{-1}&\mbox{if $h_i=(f_{\abcnj})^{-1}$ for some ${\abcnj}\in\rem{\nu}$}.\\
\end{array}\right.
\end{displaymath}
If $T$ is a set of $\fcal$-terms let $T^{[p]}=\{t^{[p]}:t\in T\}$.

Given  $p\in\detp$ ,
$\xi\in\oo$ and a  function  $F\in\bijpoo$,
if $p\rest{\xi}\force$``$p(\xi)\cup F\in Q_{\xi}$''
then write $\pand p;\xi;F;$ for  the condition $q\in\detp$ defined
by the formula
\begin{displaymath}
q(\zeta)=\left\{
\begin{array}{ll}
p(\zeta)\cup F&\mbox{if $\zeta=\xi$,}
\\
p(\zeta)&\mbox{otherwise.}
\end{array}\right.
\end{displaymath}

If  $p\rest{\xi}\in\gcal_{\xi}$, then let
$
t^{[p]}[\gcal_{\xi}]=\bigcup\{
t^{[p\land q]}:q\in\gcal_{\xi}\}
$
and 
$T^{[p]}[\gcal_{\xi}]=\{t^{[p]}[\gcal_{\xi}]:t\in T\}$.

If $a\subs\oo$ 
then,
by an abuse of notation,  we will write $t(a)$ for $t''a$ and 
$T(a)$ for $\bigcup\limits_{t\in T}t(a)$.

Assume now on the contrary that\\
$p'\force$ ``{\em $Z=\ooset {\ztil};{\alpha};$ is a sequence of pairwise
disjoint finite subsets of\\
\makebox[0.5in]{} $\oo$, $\ooset \util;{\beta};$ is a
sequence of finite sets of $\fcal$-terms such that\\
\makebox[0.5in]{}  $\ztil_{\alpha}\cap \util_{\beta}(r)\ne\empt$}
for each ${\alpha}<{\beta}<\oo$''. 

For each ${\alpha}<\oo$ choose a condition $p_{\alpha}\in\detp$,
a  set $y_{\alpha}\in\br\oo;{<\omega};$ and a finite
set of $\fcal$-terms $T_{\alpha}$ such that
$p_{\alpha}\force$``{\em $\ztil_{\alpha}={\hat y}_{\alpha}$ and
$\util_{\alpha}={\hat T}_{\alpha}$}''. We can assume that
  every $T_{\alpha}$ is closed
under subterms.

\begin{sublemma}
\label{sl:ab} There are ${\alpha}<{\beta}<\oo$ and  a condition
$p\in\detp$ such that $p\le p_{\alpha},p_{\beta}$ 
and
$p\force``{T_{\beta}^{[p]}(r)\cap y_{\alpha}=\empt}''$.
\end{sublemma}

\proof \wolog
$\{\supp(p_\alpha):\alpha<\oo\}$ forms a $\Delta$-system with
kernel $d$, $-1\in d$ and $p_\alpha\rest{d}=q$ for each $\alpha<\oo$.
Then $q(-1)\in\fin(\omega,2)$. Take $D=\dom(q(-1))$.
%there are countable ordinals ${\sigma}$ and
%$\{{\sigma}_{\alpha}:{\alpha}<{{\omega}_1}\}$ such that for each
%${\alpha}<\oo$ we have that $p_{\alpha}\rest {\xi}_{{\sigma},0}=q$
%and $\dom(p_{\alpha}\setm q)\subs {\xi}_{{\sigma}_{\alpha},0}\setm
%\bigcup\limits_{{\gamma}<{\alpha}}{\xi}_{{\sigma}_{\gamma},0}$.
%Let $D=\dom(q)\cap {\omega}$.

Consider a   $t\in T_{\omega}$. If
${\alpha}_0<{\alpha}_1<{\omega}$ then $u=(t^{[p_{{\alpha}_0}\cup
p_{{\alpha}_1}\cup p_{\omega}]})^{-1}$
is a  1--1 function,  and so
$u(y_{{\alpha}_0})\cap u(y_{{\alpha}_1})=\empt$.
Thus
\begin{displaymath}
|\{{\alpha}<{\omega}:
(t^{[p_{{\alpha}}\cup p_{\omega}]})^{-1}
(y_{\alpha})\cap D\ne\empt\}|\le |D|.
\end{displaymath}
But $T_{\omega}$ is finite, so there is an ${\alpha}<{\omega}$ such that
$T_{\omega}^{[p_{\alpha}\cup p_{\omega}]}(D)\cap y_{\alpha}=\empt$.
Take $E=(T_{\omega}^{[p_{\alpha}\cup p_{\omega}]})^{-1}y_{\alpha}$.
Then $D\cap E=\empt$. Put
$p=p_{\alpha}\land p_{\omega}\land q'$, where $q'$ is defined
by the equations  $\supp(q')=\{-1\}$ and $g'(-1)=\{\<m,0\>:m\in E\}$.
Then $p$ is determined and
$p\force$``$T_{\omega}^{[p]}(r)\cap y_{\alpha}=\empt$'',
because $(T^{[p]}_{\omega})^{-1}(y_{\alpha})\subs E$ and
$p\force$ ``$r\cap E=\empt$''.
\eproof
Take $T=(T_{\beta})^{-1}\stackrel{\mbox{\tiny def}}{=}
\{t^{-1}:t\in T_{\beta}\}$ and $y=y_{\alpha}$.
Then
$$p\force\mbox{``}
T^{[p]}(y)\cap r=\empt\mbox{\em but }
T(y)\cap r\ne\empt\mbox{''.}$$
We will find $p^*\le p$ such that 
$p^*\force$ ``$T^{[p^*]}(y)\cap r=\empt$ and
$p^*$ ``fully decides'' $T(y)$,
i.e., $p^*\force$ ``$T(y)=T^{[p^*]}(y)$'' contradicting to
$p\force$ ``$T(y)\cap r\ne\empt$''.

\begin{definition} Let $p^*\in\detp$, $s$ be an  $\fcal$-term, $S$ be a
set of  $\fcal$-terms,
${\rho}\in\oo$, $z\subs\oo$,  $\xi\in\oo$, $s=\<h_0,h_1,\ldots,h_{n-1}\>$.
\begin{enumerate}
\item We say that $p^*$  {\em fully decides $s({\rho})$}
iff there is a sequence $\<{\rho}_0,\ldots,{\rho}_m\>$
such that ${\rho}_0={\rho}$ and
$p^*\force $``$h_i(\hat{\rho}_i)=\hat{\rho}_{i+1}$''
for $i<m$ and either $m=n$ or $m<n$ and
$p^*\force $``$\hat{\rho_m}\notin\dom(h_m)$''.
\item $p^*$ {\em fully decides $S(z)$} iff it fully decides
$s({\rho})$ whenever $s\in S$ and ${\rho}\in z$.
\item Take
$
\dcd(p^*,S,z)=\{\<s,\rho\>\in S\times z:
\mbox{$p^*$ fully decides $s(\rho)$}\}.
$
\item If  $p^*\rest{\xi}\in\gcal_\xi$ then put 
$\dcd(p^*,S,z)[\gcal_\xi]=\bigcup\{\dcd(p^*\land q,S,z):q\in\gcal_\xi\}$.
\end{enumerate}
\end{definition}

If $q\in\detp$, $y\in\br\oo;<{\omega};$ and   $S$ is a set of $\fcal$-terms 
we will define the set of undecided evaluations  as follows: let
$\und{q,S,y}$ be the set of triples
$\<x,\nj,e\>$ for which there exist $s\in S$, $s=\<h_0,\ldots,h_l\>$, 
and $\eta\in y$ satisfying
 (a)--(d) below:
\begin{enumerate}\alabel
\item $x\in\oo$, $e\in\{-1,+1\}$ and $h_l={(f_{\abcnj})^e}$,
\item taking $u=\fsq h;l-1;$ we have
$u^{[q]}(\eta)=x$,
\item $x\notin\dom f^{[q]}_{\nj}$ if $e=1$,
\item  $x\notin\ran f^{[q]}_{\nj}$ if $e=-1$.
\end{enumerate}

For 
$\<\mu,i\>\in\oo\times\omega$ 
define $\undp{q,T,y,{\mu},i}$,
the set of undecided evaluations which will be decided
in step ${\mu}$, as follows:
it contains a triple $\<x,{\nj},e\>\in\und{q,T,y}$  iff
(A)--(C) below are satisfied:
\begin{enumerate}\Alabel
\item $\abcmi=\abcnjm$,
\item  $q\force$``$x\in\dom f'_{\abcmi}$'' provided $e=1$,
\item
 $q\force$``$x\in\ran f'_{\abcmi}$'' provided $e=-1$.
\end{enumerate}

Let 
$\undm{q,T,y,{\mu},i}$, the family of undecided
evaluations which remain undecided in step ${\mu}$, be  the set of triples
$\<x,{\nj},e\>\in\und{q,T,y,\mu,i}$  with
$\abcmi\ne\abcnjm$ or with $q\force$``$f'_{\abcmi}=\empt$''.

If $q\rest{\xi}\in \gcal_{\xi}$, $\xi\le\omi$,
then take
$$
\und{q,T,y}[\gcal_{\xi}]
=\bigcup\{\und{q\land q',T,y}:q'\in\gcal_{\xi}\},$$
$$
\undp{q,T,y,{\mu},i}[\gcal_{\xi}]
=\bigcup\{\undp{q\land q',T,y,{\mu},i}:q'\in\gcal_{\xi}\}$$
and
$$
\undm{q,T,y,{\mu},i}[\gcal_{\xi}]
=\bigcup\{\undm{q\land q',T,y,{\mu},i}:q'\in\gcal_{\xi}\}.$$

%\newpage
\begin{sublemma}There is a condition $p^*\le p$ which fully decides
$T(y)$ 
and $p^*\force$``${T^{[p^*]}(y)\cap r=\empt}$''.
\label{sl:main}
\end{sublemma}

\proof
We will define a finite, decreasing sequence of determined
conditions, $p_0,p_1,\ldots,p_{l^*}=p^*$ satisfying  (I) and (II) below
for each $l\le l^*$:
\begin{enumerate}\Rlabel
\item $p_l\force$``${T^{[p_l]}(y)\cap r=\empt}$'',
\item $|\dcd(p_{l-1},T,y)|<|\dcd(p_{l},T,y)|$.
\end{enumerate}
Put $p_0=p$. 
Assume that $p_l$ is constructed. 

Choose   the minimal $\<{\mu},i\>$ in the lexicographical order of
$\oo\times{\omega}$ with the property $(*)$ below: 
\begin{enumerate}
         \renewcommand{\labelenumi}{{$(*)$}}
         \renewcommand{\theenumi}{{$(*)$}}
\item   There is a $p'\in P_{\omega\mu+i}$,
$p'\le p_l\rest {(\omega{\mu}+i)}$, such that 
taking $p'_l=p'\land p_l$
we have 
 $\undp{p'_l,T,y,{\mu},i}\ne\empt$.
\end{enumerate}

To simplify our notation take $\xi=\omi$. 
Let $\gcal_l$ be a $P_{\xi}$-generic filter with $p'\in\gcal_l$.
%We will work in $V[\gcal_l]$.
By the minimality of $\<\mu,i\>$ we have
$T^{[p_l]}[\gcal_l](y)=T^{[p_l]}(y)$ and so
$$\tagg{\dag}
\mbox{$V[\gcal_l]\models$``$T^{[p_l]}[\gcal_l](y)\cap r =\empt$''.}$$
Clearly
 $$\und{p'_l,T,y}[\gcal_l]=
\undp{p'_l,T,y,{\mu},i}[\gcal_l]
\cup \undm{p'_l,T,y,{\mu},i}[\gcal_l].$$

The condition $p_{l+1}$ will be constructed by a finite induction.
Working in $V[\gcal_l]$ we will define a natural number  $o_l$ and determined
conditions
$p_{l,0}, p_{l,1},\ldots, p_{l,o_l}$
such that for all $k\le o_l$ (i)--(vi) below are satisfied:
\begin{enumerate}\rlabel
\item $p_{l,k}\rest{\xi}\in\gcal_l$,
\item $p_{l,k}\le p_{l,k-1}$
\item $p_{l,k}\rest{[\xi+1,\oo)}=p_l\rest{[\xi+1,\oo)}$,
\item $V[\gcal_l]\models$``$T^{[p_{l,k}]}[\gcal_l](y)\cap r=\empt$'',
\item $|\dcd(p_{l,k-1},T,y)[\gcal_l]|<|\dcd(p_{l,k},T,y)[\gcal_l]|$, 
\item $\undp{p_{l,o_l},T,y,{\mu},i}[\gcal_l]=\empt$.
\end{enumerate}
Let
$p_{l,0}=p_l$. Then (iv) holds by $(\dag)$.
Assume that $p_{l,k}$ is chosen.
If $\undp{p_{l,k},T,y,{\mu},i}[\gcal_l]=\empt$ then put $o_l=k$.
Let us remark that (v) and $|\dcd(p_{l,k+1},T,y)[\gcal_l]|\le |T\times y|$
imply that $o_l\le |T\times y|$.
Suppose now that (vi) fails for $k$ and   pick a witness  
$\<x_k,\<\nu_k,j_k\>,e_k\>\in\undp{p_{l,k},T,y,{\mu},i}[\gcal_l]$.
Let $T_k=T^{[p_{l,k}]}[\gcal_l]$, choose a
\begin{displaymath}
z_k\in
\left\{
\begin{array}{ll} 
\atil^{\mu}_i[\gcal_l]\setm
\left(T_k^{-1}(r\cup y)\cup T^k(r\cup y)\right)&\mbox{if $e_k=1$}\\
\btil^{\mu}_i[\gcal_l]\setm
\left(T_k^{-1}(r\cup y)\cup T^k(r\cup y)\right)&\mbox{if $e_k=-1$}
\end{array}\right.
\end{displaymath}
and take 
\begin{displaymath}
F_k=
\left\{
\begin{array}{ll}
\mbox{$\{\<x_k,z_k\>\}$}&\mbox{if $e_k=1$}\\
\mbox{$\{\<z_k,x_k\>\}$}&\mbox{if $e_k=-1$.}
\end{array}\right.
\end{displaymath}

Let us remark that $z_k$ is chosen
from a non-empty set, because both $\atil^{\mu}_i[\gcal_l]$ and 
$\btil^{\mu}_i[\gcal_l]$ are $(\fcal^i_\mu,r)$-large. 

Choose $p^*_{l,k}\in\gcal_l$ such that 
%$p^*_{l,k}\le p_{l,k}\rest{\xi}$,
$$\mbox{$p^*_{l,k}\force$``$\pand p_{l,k};\xi;F_k;\in P_{\oo}$ and
$\<x_k,\<\nu_k,j_k\>,e_k\>\in\undp{p_{l,k},T,y,{\mu},i}[\gcal_l]$''}
$$
and put  $p_{l,k+1}=\pand (p^*_{l,k}\land p_{l,k});\xi;F_k;$.
We should check only (iv).
Working in $V[\gcal_l]$ assume on the contrary that 
 there were $\alpha\in y$ and $t\in T$, $t=\fsq h;m-1;$, such that
$t^{[p_{l,k+1}]}[\gcal_l](\alpha)\in r$.
We can assume that $t$ is of minimal length.
Write $\alpha_0=\alpha$ and $\alpha_{s+1}=h_s(\alpha_s)$ for $s<m$.
Since (iv) holds for $k$, it follows that
$t^{[p_{l,k}]}[\gcal_l](\alpha)\notin r$.  
Suppose first that $e_k=1$. Then there must be an $s<m$ such that
$\alpha_s=x_k$ and $h_s=f_{\abcel}$ , or 
$\alpha_s=z_k$ and $h_s=(f_{\abcel})^{-1}$, where
$\abcelm=\abcmi$. Pick the minimal $s$. 
Then $\alpha_s\in T^{[p_{l,k}]}[\gcal_l](y)$, because $T$
is closed under subterms. But 
$z_k\notin T^{[p_{l,k}]}[\gcal_l](y)$, so $\alpha_s=x_k$ and 
$\alpha_{s+1}=z_k$. But  $t$ is of minimal length and 
$T$ is closed under subterms, so we have that there are no
$s'>s$ with $\alpha_{s'}=x_k$ and $h_{s'}=f_{\abcel}$ , or with 
$\alpha_{s'}=z_k$ and $h_{s'}=(f_{\abcel})^{-1}$.
But it means that $z_k=\alpha_{s+1}\in (T^{[p_{l,k}]}[\gcal_l])^{-1}(r)$,
which contradicts to the choice of $z_k$. If $e_k=-1$, then a similar
argument works.  
Condition (v) obviously holds by the construction.

So we have $p_{l,o_l}$. 
Choose a condition $p^*_l\in\gcal_l$ such that 
$p^*_l\le p_{l,o_l}\rest{\xi}$,
$p^*_l\force$``$t^{[p_{l,o_l}]}[\gcal_l](y)\cap r=\empt$''
and taking $p_{l+1}=p^*_l\land p_{l,o_l}$ we have
$$
\mbox{$p^*_l\force$``$t^{[p_{l,o_l}]}[\gcal_l](y)=
t^{[p_{l+1}]}(y)$''.}
$$
Since $t^{[p_{l,o_l}]}[\gcal_l](y)=t^{[p_{l+1}]}[\gcal_l](y)$,
it follows that $p_{l+1}$ satisfies (I). (II) is clear from the construction.



%We stop the construction if 
%$\undp{p_{l,o_l},T,y,{\mu},i}[\gcal_l]=\empt.$
%It must happen after no more than $|T\times y|$ steps,
%because $$\dcd(p_{l,k},T,y)<\dcd(p_{l,k+1},T,y)\le |T\times y|.$$ 
%Put $p_{l+1}=p_{l,o_l}$.

%After getting $F_o$ choose a $p_{l+1}\le p_l$ from $\detp$ with
%$p_{l+1}\rest{\omega\mu+i}\in\gcal_l$,
%$p_{l+1}(\omega\mu+i)=p_l(\omega\mu+i)\cup F_o$ and
%$p_{l+1}\rest[{\xi}_{{\mu},i+1},\oo)=
%p_l\rest[{\xi}_{{\mu},i+1},\oo)$. The induction hypothesis
%remains true by $(\dag)$ for $o$.


If there is no more  $\<\mu,i\>$ with property $(*)$, then we finish
the construction of the decreasing sequence
$p_0,p_1,\dots,p_{l^*}$.
This must happen  after at most
$|T\times y|$ steps, because (II) holds and $|\dcd(p_{l},T,y)|\le
|T\times y|$.
Let $p^*=p_{l^*}$. Since there are no more $\<\mu,i\>$
satisfying $(*)$, it follows that  $p^*$ fully decides $T(y)$ and
$p^*\force$``$T^{[p^*]}(y)\cap r=\empt$'', and so  $p^*$
satisfies the requirements of the sublemma.
\eproof

We continue the proof of  lemma \ref{lm:rfgood}. Let $\gcal_{\oo}$ be any
$P_{\oo}$-generic filter with $p^*\in\gcal$. Then in
$V[\gcal_{\oo}]$ we have $T^{[p^*]}(y)=T(y)$, because $p^*$ fully
decides $T(y)$, and  so $T(y)\cap r=\empt$. But $p\in\gcal_{\oo}$, and
$p\force$``$T(y)\cap r\ne\empt$. Contradiction, the lemma is
proved.
\eproof


\end{case}
\begin{case}{(ii)}{\ $P=\fin(\poo,2)$.}
Since the proof in this case is simpler then in case (i) and it doesn't
demand new ideas, we  only sketch it  and the details are left to the readers.

Consider the iterated forcing system
$\<P_{\nu}:{\nu}\le\poo,Q_{\nu}:{\nu}<\poo\>$ with finite support , where
$Q_0=\fin({\omega},2)$ and $Q_{\nu}=\fin(\oo,{\omega})$ for
$1\le{\nu}<\poo$. Since $P$ is isomorphic to a dense subset of
$P_{\poo}$, we must show that $\vppoo$ contains a suitable permutation group.

Let  $r$ be the Cohen real in $V^{Q_0}$ given by
the $Q_0$-generic filter.
We will define, by induction on ${\mu}$, permutation groups $G_{\mu}$
on $\oo$ for $1\le{\mu}\le\poo$ such that
\begin{enumerate}\alabel
\item $G_{\nu}\subs G_{\mu}$ for ${\nu}<{\mu}$,
\item $\vpm\models$``the pair $(G_{\mu},r)$ is good''.
\end{enumerate}
\begin{case}{1}{\ ${\mu}=1$.}
Take $G_0=\pperm\oo;V;$. Standard density arguments give that
$G_0$ satisfies (b).
\end{case}
\begin{case}{2}{\ ${\mu}$ is limit.}
Take $G_{\mu}=\bigcup\limits_{{\nu}<{\mu}}G_{\nu}$.
A suitable modification of proof of lemma \ref{lm:rfgood}
gives that $G_{\mu}$ satisfies (b).
\end{case}
\begin{case}{3}{${\mu}={\nu}+1$}
We will work in $\vpn$.
Pick sets $X_{\nu},Y_{\nu}\subs\oo$ with
$|X_{\nu}|=|\oo\setm X_{\nu}|=|Y_{\nu}|=|\oo\setm Y_{\nu}|=\oo$.
Since $(G_{\nu},r)$ is a good pair, we can fix partitions
$\{X^0_{{\nu},{\alpha}}:{\alpha}<\oo\}$,
$\{Y^0_{{\nu},{\alpha}}:{\alpha}<\oo\}$,
$\{X^1_{{\nu},{\alpha}}:{\alpha}<\oo\}$ and
$\{Y^1_{{\nu},{\alpha}}:{\alpha}<\oo\}$ of
$X_{\nu}$, $Y_{\nu}$, $\oo\setm X_{\nu}$ and $\oo\setm Y_{\nu}$,
respectively, such that each $X^i_{{\nu},{\alpha}}$ and
$Y^i_{{\nu},{\alpha}}$ are countable and $(G_{{\nu}},r)$-large.
%Let $R_{\nu}$ be the set of bijections $f$ between finite subsets
%of $\oo$ satisfying $(*)$ below:
%$$\tagg{*}
%f''X^i_{{\nu},{\alpha}}\subs Y^i_{{\nu},{\alpha}}
%\mbox{ for each ${\alpha}<\oo$ and $i\in 2$.}
%$$
Let
$$R_{\nu}=\{f\in\bijpoo:\mbox{$|f|<\omega$,  
$f''X^i_{{\nu},{\alpha}}\subs Y^i_{{\nu},{\alpha}}$
for each ${\alpha}<\oo$ and $i\in 2$
}\}.
$$ 
Take $\rcal_{\nu}=\<R_{\nu},\supseteq\>$.
Then $\rcal_{\nu}$ is isomorphic to $Q_{\nu}$, so the $Q_{\nu}$-generic
filter over $\vpn$ gives us an $\rcal_{\nu}$-generic filter $\gcal_{\nu}$
over $\vpn$. Take $g_{\nu}=\cup\gcal_{\nu}$ and let
$G_{\mu}$ be the subgroup of $\pperm\oo;V^{P_{\mu}};$
generated by $G_{\nu}\cup\{g_{\nu}\}$.
\end{case}
The proof of that  (b) remains true 
is a suitable modification of the proof of lemma \ref{lm:rfgood} and
it is left to the readers.

So $G=G_{\poo}$ is $\omega$-inhomogeneous, because the  pair $(G,r)$ is
good.
Using a bookkeeping function we can ensure that
$\{(X_{\nu},Y_{\nu}):{\nu}<\poo\}$ enumerates all the pairs
$(X,Y)$ of uncountable, co-uncountable subsets of $\oo$ in $\vppoo$. 
%satisfying $|X|=|Y|=|\oo\setm X|=|\oo\setm Y|=\oo$
 Since
$g''_{\nu}X_{\nu}=Y_{\nu}$, it follows that $G$ is $\oo$-
homogeneous.
\end{case}

%\noindent\eproof

%\input{sketch}
%\input{tranref}
\begin{thebibliography}{9}
\bibitem{H} A. Hajnal, {\em A remark on the
homogeneity of infinite permutation
groups}, preprint
\bibitem{J} T. Jech, {\em Set Theory}, Academic Press. New York, 1978
\bibitem{N} P. M. Neumann, {\em Homogeneity of infinite permutation groups},
{\sc Bull. London Math. Soc.} {\bf 20} (1988), 305-312
\bibitem{Ny} P. Nyikos, {\em Transitivities of permutation groups},
preprint
\bibitem{ST} S. Shelah, S. Thomson, {\em Homogeneity of infinite permutation
groups}, preprint
\end{thebibliography}


%\input{case2}
\end{document}
