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\title{Some Remarks on a Question of D.~H. Fremlin
Regarding $\epsilon$-Density}
\date{December 27, 1999\\
(revised February 2, 2000)}
\author{Arthur W.~Apter\\
Department of Mathematics\\
Baruch College of CUNY\\
New York NY 10010, USA\\
http://math.baruch.cuny.edu/$\sim$apter\\
awabb@cunyvm.cuny.edu\\
\\
Mirna D\v zamonja\\
Department of Mathematics\\
University of East Anglia\\
Norwich NR4 7TJ, UK\\
http://www.mth.uea.ac.uk/people/md.html\\
M.Dzamonja@uea.ac.uk}
\begin{document}
\maketitle
\begin{abstract}
We show the relative
consistency of $\aleph_1$ satisfying a combinatorial property
considered by David Fremlin (in the question DU from his list)
in certain choiceless inner models. This is demonstrated by first
proving the property is true for Ramsey cardinals. In contrast,
we show that in $ZFC$, no
cardinal of uncountable cofinality can satisfy
a similar, stronger property.
The questions considered by D.~H. Fremlin are if families of finite subsets
of $\omega_1$ satisfying a certain density condition necessarily contain
all finite subsets of an infinite subset of $\omega_1$,
and specifically if
this and a stronger property hold under $MA+\neg CH$.
Towards this we show that if $MA+\neg CH$ holds,
then for every family ${\cal A}$ of $\aleph_1$ many
infinite subsets
of $\omega_1$, one can find a family ${\cal S}$
of finite subsets of $\omega_1$ which is dense
in Fremlin's sense, and does not contain all finite subsets
of any set in ${\cal A}$.
We then pose some open problems related to
the question.
\end{abstract}
\baselineskip=24pt
\section{Introduction}\label{s1}
\begin{definition}
Let $0 < \gep \le 1$ be fixed but arbitrary,
and let $\gk$ be an infinite cardinal.
Call a family
${\cal S} \subseteq {[\gk]}^{< \aleph_0}$
$\gep$-{\em dense open} iff ${\cal S}$ is downward closed, i.e.,
$p \in {\cal S}$ and $q \subseteq p$ implies
$q \in {\cal S}$, and ${\cal S}$ in addition satisfies
the property
\[
\forall p \in {[\gk]}^{< \aleph_0}
\exists q \subseteq p\,
[q \in {\cal S} \wedge |q| \ge \gep \cdot |p|].
\]
\end{definition}
D.~H. Fremlin in
question DU from \cite{Un} asks if $\gk = \ha_1$
and we have $\gep$ and ${\cal S}$ as in the
above, then does there necessarily
exist $A \in {[\ha_1]}^{\aleph_0}$
so that
${[A]}^{< \aleph_0} \subseteq {\cal S}$? In fact this particular question is
credited to S. Argyros, while the original question of
D.~H. Fremlin is if under
$MA+\neg CH$ one can find an uncountable such $A$.
For the motivation behind the question,
we refer the reader to Fremlin's note \cite{Fr2}.
The reason one would ask the question in the above form is that an
apparently folklore
argument (communicated to us by Ilijas Farah)
shows that $CH$ implies that there
is ${\cal S}$ on $\aleph_1$ which is $1/2$-dense open, but contains
no $[A]^{<\aleph_0}$ for an uncountable
$A$, while an easy modification of an argument by P. Erd\"os and R. Rado in
\cite{ER}
answers negatively the analogue of Fremlin's question with
$\ha_0$ in place of $\ha_1$
(see the appendix for these facts).
Ilijas Farah brought to our attention the fact
that $\epsilon$-density is a variation on Kelley's intersection
number \cite{Ke} of a
family of sets, but that if in
question DU,
instead of requiring that ${\cal S}$ be
$\epsilon$-dense open,
we ask
that ${\cal S}$ be a family of finite sets obtained from some family
with the intersection number positive,
then the problem has a positive solution - there
is an infinite set with all finite subsets in ${\cal S}$, and under
$MA(\aleph_1)$ there is an uncountable such set.
Note \cite{Fr2} by D.~H. Fremlin deals with certain minimal
$\epsilon$-dense families and furthers the connection with intersection
numbers.
In Section \ref{s2}, we will define the
notions of a cardinal $\gk$ being Fremlin
and being strongly Fremlin.
We show that modulo the existence of a Ramsey cardinal, it
is consistent that $\aleph_1$ is strongly Fremlin in a model of
$ZF$ which satisfies $DC$. We then show in Section
\ref{s3} that if the original
condition in question DU is fortified, no
cardinal of uncountable cofinality
can be Fremlin in this stronger sense, in a model of $ZFC$.
In Section \ref{s4a} we show that under $MA+\neg CH$, for every family
${\cal A}$ of $\aleph_1$ elements of $[\omega_1]^{\aleph_0}$,
there is a 1/2-dense open family ${\cal S}$ of finite subsets of
$\omega_1$ such that for no $A\in {\cal A}$ do we have that
$[A]^{<\aleph_0}\subseteq {\cal S}$.
The paper ends
by giving further open problems.
\section{Choiceless universes}\label{s2}
We begin by noting that if $\gk$ is
a Ramsey cardinal, then the above
property holds for $\gk$.
Specifically, we show the following.
\begin{fact}\label{f1}
Suppose $\gk$ is a Ramsey cardinal and
${\cal S}$ is $\epsilon$-dense open for an $\epsilon\in (0,1]$.
There is then $A \in {[\gk]}^\gk$ such that
${[A]}^{< \aleph_0} \subseteq {\cal S}$.
\end{fact}
\begin{proof}
Define $f : {[\gk]}^{< \aleph_0} \to 2$ by
$f(p) = 1$ iff $p \in {\cal S}$.
Since $\gk$ is a Ramsey cardinal, let
$A \subseteq \gk$, $A \in {[\gk]}^\gk$
be such that $A$ is homogeneous for $f$.
We show that $A$ is our desired set
by considering the following two cases.
\begin{enumerate}
\item\label{c1} There are unboundedly many
$n < \omega$ so that
$f '' {[A]}^n = \{ 1 \}$, i.e., so that
${[A]}^n \subseteq {\cal S}$.
Since ${\cal S}$ is downward closed, we automatically
have that
${[A]}^{< \aleph_0} \subseteq {\cal S}$.
\item\label{c2} Case \ref{c1} doesn't hold, i.e.,
there is $n_0 < \omega$ so that for all
$n \ge n_0$, we have
$f '' {[A]}^n = \{ 0 \}$.
This means that if $n \ge n_0$ and
$p \in {[A]}^n$, then $p \not\in {\cal S}$.
In this situation, let $m > n_0$
be large enough so that
$m \cdot \gep > n_0$, and let
$p \in {[A]}^m$.
By the properties of ${\cal S}$, there must
be $q \subseteq p$, $q \in {\cal S}$ so that
$|q| \ge \gep \cdot |p|$, i.e.,
$|q| \ge m \cdot \gep
> n_0$.
As $f(q) = 1$, this contradicts
that $q \in {[A]}^\ell$ for some
$\ell > n_0$, i.e., $f(q) = 0$.
\end{enumerate}
The contradiction just obtained means that
Case \ref{c2} above can't hold, i.e., that
the homogeneous set $A$ for $f$ has the
desired properties.
This completes the proof of Fact \ref{f1}.
\end{proof}
More generally,
\begin{definition}
An
infinite cardinal $\gk$ is called
{\em Fremlin} if for every $\epsilon$-dense open
family ${\cal S}$ of finite subsets of $\kappa$,
there is an infinite $A\subseteq \kappa$ all of
whose finite subsets are contained in ${\cal S}$.
If we can
always
guarantee that such an $A$ can be found with
$|A| = \gk$, we say that $\kappa$ is
{\em strongly
Fremlin}.
\end{definition}
The proof of Fact \ref{f1} shows that a
Ramsey cardinal is strongly
Fremlin and that an
Erd\"os cardinal $\gk$,
i.e., a cardinal $\gk$ such that
$\gk \to {(\omega)}^{< \omega}$,
is Fremlin. D.~H. Fremlin in \cite{Fr2}
shows that a real-valued measurable cardinal is Fremlin.
We remark that the above proof makes no
use of the Axiom of Choice. Thus,
a model in which $\ha_1$ is a Ramsey
cardinal will be a model in which
$\aleph_1$ is strongly Fremlin.
This gives us the following theorem.
\begin{theorem}\label{t1}
Con(ZFC + $\gk$ is a Ramsey cardinal)
$\implies$
Con(ZF + DC + $\aleph_1$ is strongly Fremlin).
\end{theorem}
\setlength{\parindent}{0in}
The model used for Theorem \ref{t1}
is either Jech's model of \cite{J1}
(see also \cite{J2}, Theorem 90,
pages 476-477) or Takeuti's model
of \cite{Ta}. In both models, a Ramsey
cardinal $\gk$
(instead of a measurable cardinal)
is symmetrically collapsed
to $\ha_1$, and the Ramseyness of
$\gk$ is preserved.
\setlength{\parindent}{1.5em}
We note now that there are instances
of models in which the full Axiom of
Choice is false and in which many
successor cardinals are strongly
Fremlin.
The following two theorems provide
examples of such models.
\begin{theorem}\label{t2}
Let $V \models ``$ZFC +
There is a proper class of
supercompact cardinals''.
There is then a class partial
ordering $\FP$ and a symmetric
inner model $N \subseteq V^\FP$
such that
$N \models ``$ZF + DC + Every
successor cardinal is regular +
Every uncountable limit cardinal is singular +
The successor of every regular cardinal
is strongly Fremlin''.
\end{theorem}
\begin{theorem}\label{t3}
Let $V \models ``$ZFC +
$\gk$ is almost huge +
$B, C \subseteq \gk$ are so that
$B \cap C = \emptyset$ and
$B \cup C = \{\ga < \gk : \ga$ is
a successor ordinal$\}$''.
There are then a partial ordering
$\FP \in V$ and a symmetric inner
model $N_B \subseteq V^\FP$ of
height $\gk$ such that
$N_B \models ``$ZF + $\neg AC_\omega$ +
Every limit cardinal is singular +
If $\ga \in B$, then $\ha_\ga$ is
strongly Fremlin +
If $\gb \in C$, then $\ha_\gb$ is a singular
Rowbottom cardinal''.
\end{theorem}
The models used for Theorems \ref{t2} and
\ref{t3} are the models of \cite{A83}
and \cite{A85} respectively, in which
the relevant cardinals in $N$ and
$N_B$ are Ramsey cardinals.
Observe that an immediate corollary of
Theorem \ref{t3} is that if $C = \emptyset$,
then
$N_B \models ``$Every successor cardinal is
strongly Fremlin''.
\section{Variants on containment}\label{s3}
We have already remarked that the original question appears with a
``weak" and ``strong" version of the containment (containing all
finite subsets
of an infinite set, and containing all finite subsets of a set of size
$\kappa$). If we further strengthen the containment requirement, to
request all finite subsets of a {\em club}
subset of $\kappa$ to be contained
in our $\epsilon$-dense open family, then for any $\kappa$
of uncountable cofinality
a counterexample can be constructed just in $ZFC$.
\begin{lemma}\label{clubs}
Suppose that $\kappa\ge\aleph_1$ is regular.
Then there is a 1/2-dense open family ${\cal S}={\cal S}(\kappa)$
of finite subsets of $\kappa$ such that for no club subset $C$ of $\kappa$ do
we have
\[
[C]^{<\aleph_0}\subseteq {\cal S}.
\]
\end{lemma}
\begin{Proof} Fix $\kappa$ as in Lemma \ref{clubs}.
By a theorem of Solovay
(see Lemma 7.6 and Theorem 85 of \cite{J2}), let
$\la S_\ga : \ga < \gk \ra$ be such that
$\gk = \cup_{\ga < \gk} S_\ga$,
each $S_\ga$ is stationary, and for
$\ga < \gb < \gk$, $S_\ga \cap S_\gb = \emptyset$.
Let $m_0 \ge 1$ be fixed but arbitrary. Using
$\la S_\ga : \ga < \gk \ra$,
enumerate all elements of $[\kappa]^{m_0}$ as $\langle F'_\delta:\,\delta
<\kappa\rangle$ so that each $F\in [\kappa]^{m_0}$ appears
stationarily often. For $\delta<\kappa$ such that $F'_\delta\subseteq
\delta$, define $F_\delta=F'_\delta\cup\{\delta\}$ (otherwise, $F_\delta$
is left undefined).
Now let ${\cal S}$ be the set of all finite subsets
of $\kappa$ which do not contain any $F_\delta$ as a subset.
Clearly, ${\cal S}$ is downward closed. Let us show that it is
1/2-dense. Given $F$ a finite subset of $\kappa$, if $F$ contains no
$F_\delta$, we are done, so assume that this is not the case. Let
$\{\delta_0,\delta_1,\ldots,
\delta_{n-1}\}$ list increasingly all $\delta$ such that $F_\delta
\subseteq F$. We divide $F$ into 2 disjoint sets $A$ and $B$ as follows.
By induction on $k\le n-1$,
we decide for the elements of $F\cap [0,\delta_k)$ if they are in $A$
or in $B$.
First, let $F\cap [0,\delta_0)\subseteq B$. Suppose that
we have decided for the elements of $F\cap [0,\delta_k)$ if they are in $A$
or in $B$. If $F'_{\delta_k}$ is not a subset of $A$, let $\delta_k\in A$.
Otherwise, let $\delta_k\in B$. Let $F\cap (\delta_k,\delta_{k+1})
\subseteq B$. Also, let $F\cap
(\delta_{n-1},\kappa)\subseteq B$.
We now claim that neither $A$ nor $B$ contains any $F_\delta$. Supposing
otherwise, any such $\delta$ would have to be among
$\{\delta_0,\delta_1,\ldots,
\delta_{n-1}\}$, so let $k\le n-1$ be the first such that $F_{\delta_k}
\subseteq A$ or $F_{\delta_k}
\subseteq B$. If $F_{\delta_k}
\subseteq A$, then $F'_{\delta_k}
\subseteq A$, so $\delta_k\in F_{\delta_k}\setminus A$, a contradiction.
We similarly show that the remaining case cannot happen. So, both
$A,B\in {\cal
S}
$, and as at least one of them has size $\ge 1/2 \cdot \card{F}$,
this shows that
${\cal S}$ is 1/2-dense.
Given a club $C$ of $\kappa$, let $F^\ast$ be any element of
$[C]^{m_0}$. Let $S$ be the stationary set such that $\delta\in S
\implies F'_\delta=F^\ast$. Now let $\delta\in (C\cap S)\setminus
(\Max(F^\ast)+1)$. Clearly $F'_\delta=F^\ast\subseteq\delta$,
so in particular $F_\delta$ is defined. As $\delta\in C$ and
$F'_\delta\subseteq C$, we have that $F_\delta\subseteq C$.
Hence, $F_\delta\in [C]^{<\aleph_0}\setminus
{\cal S}$.
%$\eop_{\ref{clubs}}$
\end{Proof}
\begin{lemma}\label{sing} Suppose that $\epsilon\in (0, 1]$ and
\[
\aleph_1\le \kappa={\rm cof}(\mu).
\]
Then, there is an $\epsilon$-dense
open family ${\cal S}$ of elements of $[\mu]^{<\aleph_0}$
for which there is no club $C$ of $\mu$ with $[C]^{<\aleph_0}
\subseteq {\cal S}$ iff there is
an $\epsilon$-dense
open family ${\cal T}$ of elements of $[\kappa]^{<\aleph_0}$
for which there is no club $E$ of $\kappa$ with $[E]^{<\aleph_0}
\subseteq {\cal T}$.
\end{lemma}
\begin{Proof} Let us first fix a
continuous increasing function $f$ from
$\kappa$ to $\mu$ whose image
$C^*$ is a club in $\mu$.
Then, given ${\cal S}$ as in the Lemma, let
${\cal T}=\{f^{-1}(F):\,F\in {\cal S}\}$. It is easily checked
that ${\cal T}$ is $\epsilon$-dense open in $[\kappa]^{<\aleph_0}$.
Suppose that $E$ is a club of $\kappa$ with
$[E]^{<\aleph_0}\subseteq {\cal T}$. Then it can be seen that $C=f''E$
is a club of $\mu$ with $[C]^{<\aleph_0}\subseteq {\cal S}$.
Vice versa, given ${\cal T}$ as in the Lemma, let ${\cal S}$
consist of all sets of the form $(f''F)\cup G$ where $F\in {\cal T}$
and $G$ is a finite subset of $\mu\setminus C^\ast$. It is easy to
check that ${\cal S}$ is $\epsilon$-dense open in
$[\mu]^{<\aleph_0}$. Suppose that
$C$
is a club of $\mu$ with $[C]^{<\aleph_0}\subseteq {\cal S}$.
Without loss of generality, $C\subseteq C^\ast$. Hence,
$[C]^{<\aleph_0}\subseteq \{f''F : \,F\in {\cal T}\}$, and
so letting $E=f^{-1}(C)$, we obtain a club of $\kappa$ with
$[E]^{<\aleph_0}\subseteq {\cal T}$.
\end{Proof}
\begin{theorem}\label{clubsing}
Suppose that $\lambda$ is a cardinal of uncountable cofinality.
Then there is a 1/2-dense open family ${\cal S}={\cal S}(\lambda)$
of finite subsets of $\lambda$ such that for no club subset $C$ of $\lambda$ do
we have
\[
[C]^{<\aleph_0}\subseteq {\cal S}.
\]
\end{theorem}
\begin{Proof}
If $\lambda$ is regular, the conclusion follows from Lemma \ref{clubs}.
Otherwise, let $\kappa$ be the cofinality of $\lambda$.
As in
the proof of Lemma \ref{sing}, ${\cal S}(\kappa)$ gives rise to
${\cal S}(\lambda)$ as required.
\end{Proof}
\section{An $MA$ result}\label{s4a}
\begin{theorem}\label{MA} Assume $MA$ (or even just $MA$ for Knaster property
forcing and $\ha_1$ dense sets),
and $\neg CH$. Then, for any family ${\cal A}$ of $\aleph_1$
many elements of $[\omega_1]^{\aleph_0}$, there is a 1/2-dense open
family ${\cal S}$ such that for no $A\in {\cal A}$ do we have
$[A]^{<\aleph_0}\subseteq {\cal S}$.
\end{theorem}
\begin{Proof} Let us assume the axioms from the statement of the Theorem, and
suppose that ${\cal A}$ as there is given. Let
%$\{A_\zeta:\,\zeta<\omega_1\}$
$\la A_\zeta : \zeta < \omega_1 \ra$
be a 1-1 enumeration of ${\cal A}$.
Let
$m_0 \ge 2$ be a fixed but arbitrary integer, and let
$[\omega_1]^{m_0}=
\la F_\ga : \ga < \omega_1 \ra$
%\{F_\alpha:\,\alpha < \omega_1\}$
be a 1-1 enumeration. We define
\begin{equation*}
{\mathbb P}=\left\{
\begin{split}
p:\,&\mbox{(i)}\, p\mbox{ is a finite function from
}\omega_1\mbox{ into }\omega_1\\
&\mbox{(ii)}\, p(\zeta)=\alpha\implies F_\alpha\subseteq A_\zeta\\
&\mbox{(iii)}\, \zeta\in \Dom(p)\implies [p(\zeta)\in \Ran(p\rest \zeta)
\vee
\Max(F_{p(\zeta)})\notin \cup_{\xi<\zeta,\xi\in\Dom(p)}F_{p(\xi)}]\\
&\mbox{(iv)}\, {\cal S}_p\deq\{F\in
[\omega_1]^{<\aleph_0}:\,\{\alpha:\,F_\alpha\subseteq F\,\,\&\,\,\alpha\in
\Ran(p)\}=\emptyset\}\mbox{ is }1/2\mbox{-dense}
\end{split}
\right\}
\end{equation*}
and let ${\mathbb P}$ be ordered by inclusion, i.e. $q$ is stronger than $p$,
written as $p\le q$, iff $p\subseteq q$.
\begin{Claim}\label{density} For every $\zeta<\omega_1$, the set
\[
{\cal D}_\zeta\deq\{p\in {\mathbb P}:\,\zeta\in \Dom(p)\}
\]
is dense in ${\mathbb P}$.
\end{Claim}
\begin{Proof} Suppose we are
given $\zeta<\omega_1$ and $p\notin {\cal D}_\zeta$.
Let $F^\ast\in [A_\zeta]^{m_0}$ be disjoint from $\cup_{\alpha\in \Ran(p)}
F_\alpha$,
which exists as $A_\zeta$ is infinite. Let $\beta$ be such that
$F^\ast=F_\beta$ and let $q=p\cup\{(\zeta,\beta)\}$. Clearly $p\subseteq q$ and
properties (i)--(iii) from the definition of ${\mathbb P}$ are satisfied.
Let us verify (iv).
Let $F\in [\omega_1]^{<\aleph_0}$ be given.
We can assume that
$F_\beta\subseteq F$, as otherwise the
situation is trivial by our assumptions.
Let $i\neq j\in F_\beta$. Let $F'\subseteq F\setminus\{i,j\}$ be in ${\cal
S}_p$ and such that $F'\ge[\card{F}-2]/2$, which exists as $p\in {\mathbb P}$.
Let $F''\deq F'\cup\{i\}$. Note that $F_\beta\nsubseteq F''$ as $j\notin F''$,
and that adding $\{i\}$ to $F'$ does not change the membership in ${\cal S}_p$,
as $i$ is not an element of any $F_\alpha$ for $\alpha\in \Ran(p)$. Hence
$F''\subseteq F$ is an element of ${\cal S}_q$, and $\card{F''}\ge \card{F}/2$.
%$\eop_{\ref{density}}$
\end{Proof}
\begin{Main Claim}\label{main} ${\mathbb P}$ has the Knaster property.
\end{Main Claim}
\begin{Proof} Let $\{p_i:\,i<\omega_1\}\subseteq {\mathbb P}$
be given. For $i<\omega_1$, let $D_i\deq\Dom(p_i)$.
{\noindent (1)} Without loss of generality, we can assume that
the $D_i$'s form a
$\Delta$-system with root $D^\ast$, and $\card{D_i\setminus D^\ast}=n^\ast$
is constant.
{\noindent (2)} As for every $\zeta\in D^\ast$ and $i<\omega_1$, we have
$p_i(\zeta)\in\{\alpha:\,F_\alpha\subseteq A_\zeta\}$, which is a countable
set, we may assume that $p_i\rest D^\ast=p^\ast$ is a constant.
Now the situation of $n^\ast=0$ becomes trivial, so we can assume that
$n^\ast>0$.
{\noindent (3)} Using the fact that each $D_i\subseteq \omega_1$,
we can also assume that the $\Delta$-system formed by the
$D_i$'s is head-tail-tail,
i.e., if $i\Max(\cup\{F_\alpha:\,\alpha\in \Ran(p_i)\}).
\]
{\noindent (6)} Thinning again if we have to, we may assume that for every
$k0$ and every $\epsilon$-dense
open family ${\cal S}$ of subsets of $\kappa$, there is an $S\in
[\kappa]^\lambda$ with $[S]^{<\aleph_0}\subseteq{\cal S}$. Then,
using
an appropriate variant of Fremlin's theorem above, one can prove that if
$\gk$ is singular,
$\kappa > \lambda\ge\cf(\lambda)>
\cf(\kappa)$, and $\{\theta:\,\theta\mbox{ is not }\lambda\mbox{-Fremlin}\}$
is cofinal in $\kappa$, then $\kappa$ is not $\lambda$-Fremlin.
Namely, fixing $\lambda$, we prove this by induction on $\cf(\kappa)$.
We fix a strictly increasing sequence $\langle \theta_i:\, i<\cf(\kappa)
\rangle$ of cardinals which are not $\lambda$-Fremlin and increase to
$\kappa$. By the inductive hypothesis we can assume that the
sequence is continuous. By Fremlin's theorem we can find
for each $i$ a 1/2-dense open ${\cal S}_i\subseteq [\theta_i]^{<\aleph_0}$
which demonstrates that $\theta_i$ is not $\lambda$-Fremlin.
Considering
\[
{\cal S}_i\cup [{\cal S}_{i+1} \cap[[\theta_i,\theta_{i+1})]^{<\aleph_0}]
\cup\{A\cup B:\,A\in {\cal S}_i, B\in {\cal S}_{i+1}
\cap[[\theta_i,\theta_{i+1})]^{<\aleph_0}\}
\]
we can see that, by changing $\langle {\cal S}_i:\,i<\cf(\kappa)\rangle$
inductively, we can assume that ${\cal S}_{i+1}\cap [\theta_i]^{<\aleph_0}
={\cal S}_i$, and it is also clear that for $i$ limit we can assume that
${\cal S}_i=\cup_{j*0$ and ${\cal S}\subseteq
[\kappa]^{<\aleph_0}$ is $\epsilon$-dense open, is there necessarily a finitely
additive measure $\nu:\,{\cal P}(S)\longrightarrow[0,1]$ such that
$\nu(\{I:\,\xi\in I\in {\cal S}\})>0$ for all $\xi<\kappa$?
\end{Question}
\section{Appendix}\label{s5}
We comment on two facts mentioned earlier in the paper.
\begin{fact}\label{omega} $\aleph_0$ is not Fremlin.
\end{fact}
\begin{Proof} This is a simple modification of the Erd\"os-Rado
argument \cite{ER}
that $\aleph_0$ is not Ramsey. We just consider
the family of all finite subsets of $\omega$ whose smallest
element is at least as large as its cardinality.
%$\eop_{\ref{omega}}$
\end{Proof}
There is a paper by P. Pudlak and V. R\"odl \cite{PR}
in which the authors
classify all the families ${\cal S}$ of finite subsets of $\omega$ which are
downward closed, every infinite subset of
$\omega$ has an initial segment in ${\cal S}$, yet no infinite subset
of $\omega$ has all initial segments in ${\cal S}$.
\begin{fact}\label{CH} If $CH$ holds, then $\aleph_1$ is not
strongly Fremlin.
\end{fact}
\begin{Proof}
This is the apparently folklore argument
mentioned in Section \ref{s1} and communicated
to us by Ilijas Farah.
By $CH$, fix an increasing sequence
$\langle A_\xi:\,\xi<\omega_1\rangle$ of null sets which cover
the unit interval. For every $\xi$,
fix a compact set $K_\xi$
of measure 1/2 which is disjoint from $A_\xi$. Define ${\cal S}$
to be the family of those finite subsets $F$ of $\omega_1$
for which $\bigcap\{K_\xi:\,\xi\in F\}\neq\emptyset$. Clearly,
${\cal S}$ is closed under taking subsets. If $\card{F}\ge 2n$,
then $F$ clearly has a subset of size at least $n$ which is in
${\cal S}$ - namely consider $\int\Sigma_{\xi\in F}\chi_{K_\xi}$,
and apply the Mean Value Theorem.
On the other hand, any $X\subseteq \omega_1$ satisfies $[X]^{<\aleph_0}
\subseteq {\cal S}$ iff $\bigcap\{K_\xi:\,\xi\in X\}\neq\emptyset$,
by compactness. If such $X$ were to be uncountable, we would have
a contradiction with the choice of the $A_\xi$'s.
%$\eop_{\ref{CH}}$
\end{Proof}
%In the above example, there is always an infinite set $X$ with the
%required property, and adding a random real adds an uncountable
%set $X$ with the required property.
It would be interesting to know if one can generalize the above proof
to show that if $2^\kappa=\kappa^+$, then $\kappa^+$ is not strongly
Fremlin. The obvious analogue runs into difficulty, as we would need
$\kappa$-additivity of the measure we use.
\setlength{\parindent}{0in}
\bigskip
\bigskip
{\bf Acknowledgement}:
The authors would like to thank the
Department of Mathematics of the
University of East Anglia for the support
offered during the workshop in set theory
held January 12 and 13, 1999 at the
University of East Anglia and to NATO for support
through their grant number PST.CLG975324.
In addition, we are both very grateful to
Ilijas Farah for
telling us the problem and for his many
helpful comments and suggestions,
to David Fremlin for his
many helpful comments
and suggestions (including outlines
of alternative proofs),
and to Andreas Blass for his careful reading of
the manuscript and his many helpful comments,
suggestions, and corrections, which have been
incorporated into this version of the paper.
\eject
\setlength{\parindent}{1.5em}
\begin{thebibliography}{99}
\bibitem{A83} A.~Apter, ``Some Results on
Consecutive Large Cardinals'',
{\it Annals of Pure and Applied Logic 25},
1983, 1--17.
\bibitem{A85} A.~Apter, ``Some Results on
Consecutive Large Cardinals II: Applications
of Radin Forcing'',
{\it Israel Journal of Mathematics 52},
1985, 273--292.
\bibitem{ER} P.~Erd\"os and R.~Rado, ``Combinatorial Theorem
on Classifications of Subsets of a Given Set", {\it Proceedings
of the London Mathematical Society 3}, 1952, 417--439.
\bibitem{GaPr} F.~Galvin and K.~Prikry, ``On Kelley's Intersection
Numbers", to appear in the
{\it Proceedings of the American Mathematical Society}.
\bibitem{Un} D.~H.~Fremlin, ``A List of Open Problems",
the ftp site of the author ftp.essex.ac.uk/pub.
\bibitem{Fr2} D.~H.~Fremlin, ``A Note on Problem DU",
preprint October 1998, June 1999 and
the ftp site of the author ftp.essex.ac.uk/pub/measuretheory/probDU.tex.
\bibitem{J1} T.~Jech, ``$\omega_1$ can be
Measurable'', {\it Israel Journal of Mathematics 6},
1968, 363--367.
\bibitem{J2} T.~Jech, {\it Set Theory}, Academic Press,
New York, San Francisco, and London, 1978.
\bibitem{Ke} J.~L.~Kelley, ``Measures on Boolean
Algebras", {\it Pacific Journal of
Mathematics 9}, 1959, 1165--1177.
\bibitem{PR} P.~Pudlak and V.~R\"odl,
``Partition Theorems for Systems of Finite
Subsets of Integers",
{\it Discrete Mathematics 39}, 1982,
67--73.
\bibitem{Ta} G.~Takeuti, ``A Relativization of Axioms of
Strong Infinity to $\omega_1$'',
{\it Annals of the Japanese Association of
Philosophy and Science 3}, 1970, 191--204.
\end{thebibliography}
\end{document}
*