% Some interesting problems
% A. Miller miller@math.wisc.edu
% LaTeX 2e
\def\revised{April 2015}
\def\note{ This is an update of my problem list \cite{m31}. Problems not in
\cite{m31} have a ``$*$''. I tried to include as many references as I could
think of. If you know anything about these problems or could supply any
missing references or corrections (missing attributions or misattributions),
please let me know. The latest version is kept at:
{http://www.math.wisc.edu/$\sim$miller} }
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\begin{document}
\begin{center}
{\large Some interesting problems}\\
Arnold W. Miller\footnote{\note}\\
\revised
\end{center}
\topic{Analytic sets}
% 1.1
\prob (Mauldin) Is there a $\Sigma^1_1$ set X universal for
$\Sigma^1_1$ sets which are not Borel? Suppose $B\in\Sigma^1_1$ and
for every Borel A, $A\leq_W B$. Does this imply that for every
$\Sigma^1_1$ A, $A\leq_W B$. (This refers to Wadge reducible.)
\answer{The first question was answered by Hjorth \cite{hjorth1}
who showed that it is independent.}
% 1.2
\prob A subset $A\subset\bsp$ is compactly-$\Gamma$ iff for every
compact $K\subset\bsp$ we have that $A\cap K$ is in $\Gamma$.
Is it consistent relative to ZFC that compactly-$\Sigma^1_1$ implies
$\Sigma^1_1$? (see Miller-Kunen \cite{m17}, Becker \cite{beckanal})
% 1.3
\prob (Miller \cite{m17})
Does $\Delta^1_1$ = compactly-$\Delta^1_1$ imply
$\Sigma^1_1$ = compactly-$\Sigma^1_1$?
% 1.4
\prob (Prikry see \cite{fried}) Can $L\cap\bsp$ be a nontrivial $\Sigma^1_1$
set? Can there be a nontrivial perfect set of constructible reals?
\answer{No, for first question Velickovic-Woodin \cite{vw}.
No, for second question
Groszek-Slaman \cite{grosla}. See also Gitik
and Golshani \cite{gitik,gitik2}.}
% 1.5
\newprob (A.Ostaszewski, email 9-92)
Consider Telgarsky's game $G(T)$ where $T\subseteq \csp$.
Player I plays a countable cover of $T$
Player II chooses one- say $X_n$.
\par Player I wins iff $\cap\{cl(X_n):n\in\omega\}\subseteq T$.
\par \noindent It is known that
\par (a) Player I has winning strategy iff T is analytic.
\par (b) If there exists $A$ an analytic subset of $cl(T)$ not Borel
separated from $T$, then Player II has a winning strategy.
\par\noindent Is the converse of (b) true?
% 1.6
\newprob Does there exists an analytic set which is not
Borel modulo Ramsey null? Same question for the ideal generated
by closed measure zero sets. For measure
see Grzegorek and Ryll-Nardzewski \cite{GR}.
\answer{Yes for second question, Mauldin \cite{maulbest}.
Dodos \cite{dodos} gives an example of
of an analytic subset $A$ of $[\omega]^\omega$
for which there does not exist a Borel superset
$B$ of $A$ such that the difference $B\setminus A$
is Ramsey-null.}
% 1.7
\newprob (Sierpinski \cite{sier70}) Does there exists
a set of reals $E$ such that every (uncountable) analytic
set of reals is the one-to-one continuous image of $E$?
\answer{Yes, Slaman \cite{slasier}. Earlier version of this problem
had a misprint in it due to my faulty French.}
% 1.8
\newprob (Jockusch conversation 10-95)
Let for $A\subseteq \omega$ let $D(A)=\{a-b:a,b\in A\}$. Is the set
$\{D(A): A\subseteq\omega\}$ Borel?
\answer{No, Schmerl \cite{schmerl}.}
% 1.9
\newprob Suppose $I$ is a $\sigma$-ideal generated by
its $\Pi^0_2$ members.
Then is it true that for any analytic set $A$
either $A\in I$ or $A$ contains a $\Pi^0_3$ set not in I?
This is suggested by a theorem of Solecki \cite{solec}
that says that for any
$\sigma$-ideal $I$ generated its closed members and analytic set $A$,
either $A\in I$ or $A$ contains a $G_\delta$ set not in I.
\topic{Axiom of Determinacy}
% 2.1
\prob Does AD imply that $2^{\omega_1}$ is the $\omega_1$ union of meager
sets? \answer{Yes, Becker \cite{becker}.}
% 2.2
\prob Does AD imply that there does not exist $\omega_2$ distinct
$\Sigma_2^1$ sets?
\answer{Yes, Hjorth \cite{hjorthbsl}.} % hjorth6
% 2.3
\prob Is there a hierarchy of $\Delta^1_2$ sets?
% 2.4
\prob Does AD imply every set is Ramsey?
\answer{Yes, if also assume $V=L[\reals]$ for references
see Kanamori \cite{kan} page 382. Yes for AD$_R$, see
Prikry \cite{prikram}.}
% 2.5
\prob (V. Delfino \cite{cab}) (Conjecture) If $f:\csp\to\csp$ is
Turing invariant ($x\equiv_T y\to f(x)\equiv_T f(y)$) then
there exists $z$ such that either for every
$x\geq_{T} z\;\;f(x)\geq_{T} x$ or there exist $c$ such that
for every $x\geq_{T} z\;\;f(x)\equiv_T c$.
% 2.6
\newprob (the last Victoria Delfino problem \cite{delfino} or see
Hauser \cite{haus})
Does ZFC + projective uniformization + every projective set
has the Baire property and is Lebesgue measurable prove
projective determinacy?
\par Woodin has shown that ZFC + projective uniformization +
every projective set
has the Baire property and is Lebesgue measurable implies
that $x^\dagger$ exists for every real $x$.
\answer{Steel has shown that the answer is no.
see Schindler \cite{schindler}}
\topic{Combinatorial cardinals less than the continuum}
% 3.1
\prob (van Douwen \cite{vandou})
If every $\omega_2$ descending sequence in
$P(\omega)/{\mbox{finite}}$
has something beneath it is it true that every family of $\omega_2$
sets with the IFIP has something beneath it? (does $\mathfrak t=\mathfrak p$).
\answer{ $\mathfrak t=\mathfrak p$ was proved by Malliaris and Shelah
\cite{malshel}.}
% 3.2
\prob (Hechler \cite{hech}) Let M be a countable transitive model of ZFC.
Does there exists a generic extension $M[f_n : n\in\omega]$ with
$f_n\in\bsp$ such that
$f_n$ eventually dominates every element of
$M[f_m : m>n]\cap\bsp$?
\par For something similar with Sacks forcing see Groszek \cite{gros}
and Kanovei \cite{kanovei}.
\answer{No, Hjorth (email 1996)
see Brendle \cite{brendle} Theorem 0.5.}
% 3.3
\prob (Dow) Does the following imply ${\mathfrak p}\geq\kappa$:
$\forall X,Y\subset\infsets,|X|,|Y|<\kappa$ and $\forall A\in X,B\in Y$
$A\cap B$ finite, there exists $U\subset\omega$ such that
for all $V\in (X\cup Y)$
\par\centerline{ $(U\cap V)$ is infinite iff $V\in X$ }
\answer{No, Dow \cite{dow33}. See also, Brendle \cite{brendq}.}
% 3.4
\prob Can the least $\kappa$ such that Indep($\kappa$) fails
have cofinality $\omega$? Indep($\kappa$) means that every
family B of $\kappa$ infinite subsets of $\omega$ there exists
an infinite subset Z of $\omega$ such that for every $A\in B$,
$|Z\cap A|=|Z\setminus A|=\omega$ (see Miller \cite{m11}).
Brendle \cite{bremad} shows that it is consistent that
the smallest MAD family can have size $\aleph_{\omega}$.
% 3.5
\prob (Kunen \cite{kunma}) Let $\mathfrak m$ be the smallest cardinal for
which MA$_{\mathfrak m}$ fails. Can we have
$\omega_2=cof(\mathfrak m)<\mathfrak m$?
% 3.6
\newprob (Scheepers 7-91, Dordal \cite{dordal})
Is it consistent that $\aleph_\omega$
embeds into $(\bsp, \leq^*)$ but not $\aleph_{\omega+1}$?
\answer{Yes, Farah \cite{farah}, also the appendix of
Cummings, Scheepers, and Shelah
\cite{css}.}
% 3.7
\newprob (Vojtas \cite{vojtas} see Vaughan \cite{vaughan})
Does ${\mathfrak r}_\sigma={\mathfrak r}$? This stands for
reaping number.
$${\mathfrak r}=\min\{|R|:\;\; R\subseteq[\omega]^\omega,
\forall X\subseteq\omega
\exists Y\in R\; Y\subseteq X \mbox{ or }
Y\subseteq (\omega\setminus X) \}$$
$${\mathfrak r}_\sigma=\min\{|R|:\;\; R\subseteq[\omega]^\omega,
\forall (X_n\subseteq \omega:\;\;n\in\omega)
\exists Y\in R \forall n \;
Y\subseteq^* X_n \mbox{ or } Y\subseteq^* (\omega\setminus X_n)
\}$$
There is also an analogous problem for the splitting cardinal
$\mathfrak s$ due to Malyhin, see Kamburelis and Weglorz \cite{kw}.
\topic{MAD families}
% 4.1
\prob (Roitman) Is it consistent that
every maximal almost disjoint
family in $\infsets$ has cardinality greater than $\omega_1$, but
there exists a dominating family F in $\bsp$ of cardinality
$\omega_1$? For a related result see Shelah \cite{shelboulder}
and also Brendle \cite{brenmob} and Hrusak \cite{hrusak}.
\answer{Yes, if we replace $\omega_1$ with
$\omega_2$, Shelah \cite{shelah700}.}
% 4.2
\prob (van Douwen) CH implies there exist $F\subseteq\bsp$ which
is maximal with respect to eventually different functions
which is also maximal
with respect to infinite partial functions also.
Is there always such a
one? What is the cardinality of the smallest?
\par This problem is discussed in Zhang \cite{zhang}
see also Zhang \cite{zhang3} for some related problems.
\answer{Dilip \cite{dilip} proves that van Douwen MAD families
exist in ZFC.}
% 4.3
\prob (Cook, Watson) Consider paths in $\omega\times\omega$. CH implies
there is a MAD family of paths. Is there always one?
\answer{No, Steprans \cite{step},
still open for dimensions $\geq 3$.}
% 4.4
\prob (Milliken, ?Hechler) A maximal almost disjoint family $X$
is a separating family iff for all $Q\in\infsets$
$$\{Q\cap P{\;\;|\;\;} P\in X \mbox{ and } |Q\cap P|=\omega \}$$
has size continuum or is finite. Are there always
separating families?
% 4.5
\prob (Erd\"{o}s, Hechler \cite{erdhech})
Does MA plus the continuum is larger than $\aleph_{\omega+2}$ imply
that there is no mad family on $\aleph_{\omega}$ of size
$\aleph_{\omega+1}$?
\answer{Settled? by Kojman, Kubis, Shelah \cite{kks}.}
% 4.6
\prob (Kunen) Call $I\subseteq\infsets$ an independent splitting family
if I is independent ( every finite boolean combination of elements of I
is infinite) and splitting ( for every $f:I\to 2$ there does not
exist an infinite X such that for every $A\in I$, $X\subset^* A^{f(A)}$,
where $A^0=A$ and $A^1=\omega\setminus A$.)
If CH or MA then there does exists an independent splitting family.
In ZFC is there one?
\answer{Yes, P.Simon \cite{simon} (Bell pointed this out),
also solved by Shelah and Brendle each independently.}
% 4.7
\prob (Fleissner) If there is a Luzin set, then is there a MAD family
of size $\omega_1$?
% 4.8
\newprob (Erdos-Shelah \cite{erdshel})
Does there always exist a
completely separable mad family?
A mad family ${\cal M}$
is completely separable iff for every $A\subseteq \omega$
if there are infinitely many $M\in {\cal M}$ which meet
$A$ in an infinite set, then there exists $M\in {\cal M}$
with $M\subseteq A$. I don't know what the relationship
of this question is to 4.4. See P.Simon \cite{psimon} also.
\topic{Forcing}
% 5.1
\prob (S. Friedman, R. David) Let $P_n=2^{<\omega_n}$. Does
forcing with $\Pi_{n\in\omega}P_n$ add a Cohen
subset of $\omega_{\omega+1}$?
\answer{Yes, Shelah \cite{shel1}.}
% 5.2
\prob (Kunen) Force with perfect $P\subset\csp$ such that
for every $I\in\infsets$ $\pi_I:P\to 2^I$ does not have a countable
range. Is $\omega_1$ collapsed?
% 5.3
\prob (van Douwen, Fleissner) Is it consistent with not
CH that for P a c.c.c partial order of
size continuum there exists a sequence $G_{\alpha}$ for $\alpha<\omega_1$
of P-filters such that for every dense set $D\subseteq P$ all but countably
many of the $G_{\alpha}$ meet D.
\answer{No, Todorcevic \cite{tod}.}
% 5.4
\prob Is there a Truss-like characterization of eventually different reals?
How about infinitely equal reals? (Truss \cite{truss} proved that
if $f$ dominates $\bsp\cap M$ and $g\in\bsp$ is Cohen over $M[f]$,
then $f+g$ is Hechler generic over $M$.)
% 5.5
\prob (Kunen \cite{kun}) Does there exists an $\omega_1$ saturated
$\sigma$-ideal
in the Borel subset of $\csp$ which is invariant under homeomorphisms
induced by permutations of $\omega$ and different from the meager ideal,
measure zero ideal, and the intersection ideal?
\answer{Partial Kechris-Solecki \cite{kecsol}. Yes,
Roslanowski-Shelah \cite{628}.}
% 5.6
\prob (van Mill) Is it consistent that every c.c.c. boolean algebra
which can be embedded
into $P(\omega)/{\mbox{finite}}$ is $\sigma$-centered?
\answer{No, M.Bell \cite{bell}, Shelah \cite{shelsous2}
gives a Borel example.}
% 5.7
\newprob Suppose $M\subseteq M[f]$ are models of ZFC and
for every $g\in\bsp\cap M$ there exists infinitely
many $n\in\omega$ such that $g(n)=f(n)$.
Must there exists a real $x\in M[f]$ which is Cohen over
$M$? (If there are two such infinitely equal reals (iteratively),
then there must be a Cohen real, see \cite{m11} and \cite{slalom}.)
% 5.8
\newprob (S.Watson, conversation with A.Dow Feb 1995)
Can a poset change its cofinality in a generic extension but no
cardinal changes its cofinality?
\answer{See Gitik \cite{gitikcof}}
\topic{Measure theory}
% 6.1
\prob (Mauldin, Grzegorek) Is it consistent that the continuum is RVM
and all sets of reals of cardinality $\omega_2$ have zero measure?
\answer{No, apparently from the Gitik-Shelah
Theorem (see Fremlin \cite{fremreal} 6F) it was deduced by
Prikry and Solovay that if $\kappa$ is
real-valued measurable, then there are Sierpinski sets of all cardinalities
less than $\kappa$.}
% 6.2
\prob (Fremlin) Can the cardinality of the
least cover of the real line by measure zero
sets have countable cofinality?
\answer{Yes, Shelah \cite{shel2}.}
% 6.3
\prob (Erd\"{o}s) For every sequence converging to zero does there exist
a set of positive measure which does not contain a similar sequence?
Falconer \cite{falc}
has shown that if the sequence converges slowly enough there
does exist such a set of positive measure. H.I. Miller \cite{him}
has shown the analogous statement for Baire category to be false.
I showed that for every sequence there exist a partition
of the reals into two sets neither of which contains a sequence similar
to the given one. See survey Svetic \cite{svetic}.
% 6.4
\prob (Erd\"{o}s) Suppose for every $n\in\omega$ the set
$A\cap [n,\infty)$ has positive measure. Must $A$ contain arbitrarily
long arithmetic progressions?
\answer{Several mathematicians have pointed out this is trivial.
Probably I misquoted Erd\"{o}s. I scribbled it down after one
of his talks when the universe was younger.
To quote Just \cite{just13-1}:
``The answer to 6.4 seems to be trivially
`yes', unless you want the differences to be integers; then the answer seems
to be trivially `no', unless you want the measure to be positive in
EVERY interval, in which case
the answer may not be so trivial. So, what should the
problem really look like?''
}
% HKase Haseo Ki and Just inquire about this May Dec 93.
% 6.5
\prob Is it possible to have a Loeb-Sierpinski set of cardinality greater
than $\omega_1$? See Leth-Keisler-Kunen Miller \cite{m25} and
Miller \cite{m27}.
% 6.6
\newprob (Louveau) If a subset A of the plane has positive measure
and contains the diagonal, then
does there exist a set B in the line of positive outer measure such than
$B^2$ is a subset of A?
\answer{According to Burke \cite{bur}, Fremlin and Shelah proved
this fails in the Cohen real model. See also Steprans \cite{stepcohen}
\S 4.4.}
\topic{Borel hierarchies}
% 7.1
\prob Is it consistent that for every countable ordinal $\alpha$ there
exists a $\Pi^1_1$ set of Baire order $\alpha$?
See Miller \cite{m1}.
% 7.2
\prob Is it consistent that for every uncountable separable metric space
X there exists a X-projective set not Borel in X? See Miller
\cite{m9},\cite{m28}.
% 7.3
\prob Is it consistent that the set of all Baire orders is the same
as the set of even ordinals $\leq\omega_1$? See Miller \cite{m33}.
% 7.4
\prob Is it true that if X is a $Q_{\alpha}$-set and Y is a $Q_{\beta}$-set
and $2\leq\alpha<\beta$ then $|X|<|Y|$? \cite{m1}
% 7.5
\prob Does $\rect^{\omega_2}_{\omega_1}=P(\omega_2\times\omega_2)$ and
$2^{\omega}=\omega_2$ imply that
$2^{\omega_1}=\omega_2$? ($\rect^{\omega_2}$ is the family of abstract
rectangles in $\omega_2\times\omega_2$ and the lower subscript is the
level of the Borel hierarchy.)
% 7.6
\prob Does $\rect^{\omega_2}_{\omega}=P(\omega_2\times\omega_2)$ imply that
for some $n<\omega\; \rect^{\omega_2}_n=P(\omega_2\times\omega_2)$?
% 7.7
\prob Does $|X|=\omega_1$ imply that X is not a $Q_{\omega}$-set?
% 7.8
\prob (Mauldin) Is it consistent that there exists a separable metric space
X of Baire order less than $\omega_1$ (i.e. for some $\alpha<\omega_1$
every Borel subset of X is $\Sigma^0_{\alpha}$ in $X$) but not every
relatively analytic set is relatively Borel?
\answer{Yes, Miller \cite{relanal}.}
% 7.9
\prob Can the Borel hierarchy on cubes in $\rect^3$ behave differently than
the Borel hierarchy on rectangles in $\rect^2$?
% 7.10
\prob (Ulam \cite{ulam}) Is there a separable metric space of
each projective
class order? ($\Sigma^1_2$-forcing?) See Miller \cite{m9},\cite{m28}.
% 7.11
\prob In the Cohen real model is there an uncountable separable metric
space of Baire order 2? In the random real model are there any separable
metric spaces of Baire order between 2 and $\omega_1$?
\answer{Answered in Miller \cite{m33}.}
% 7.12
\prob What can we say about hierarchy orders involving
difference hierarchies or even abstract $\omega$-boolean operations?
% 7.13
\prob (Stone) Is it consistent to have a Borel map $f:X\to Y$ where
$X$ and $Y$ are metric spaces and $f$ has the property that there is
no bound less than $\omega_1$ on the Borel complexity of $f^{-1}(U)$ for
$U\subseteq Y$ open? Fleissner \cite{fleiss}
shows that it is consistent there is no such $f$ using
a supercompact. See also
Fremlin, Hansell and Junnila \cite{FHJ}.
% 7.14
\prob (Ciesielski-Galvin \cite{cg}) Let $P_2(\kappa)$ be the family of
all cylinder
sets in $\kappa^3$ (where cylinder means $A\times B$ where $A\subseteq \kappa$
and $B\subseteq \kappa^2$ or anything that could be obtained like this by
permuting the three coordinates.) Is it consistent that the $\sigma$-algebra
generated by $P_2(\mathfrak c^{++})$ is equal to all subsets of
$(\mathfrak c^{++})^3$?
% 7.15
\prob (Ciesielski) Suppose every subset of $\omega_2\times\omega_2$
is in the $\sigma$-algebra generated by the abstract rectangles. Does this
continue to hold after adding $\omega_1$-Cohen reals?
% 7.16
\newprob (Fleissner \cite{fleisq})
If $X$ is a Q-set of size $\omega_1$, then is $X^2$ a Q-set?
(Not necessarily true for $X$ of cardinality $\omega_2$.)
% 7.17
\newprob (Z.Balogh, conversation March 1996)
Is it possible to have $H\subseteq P(\reals)$ such that
the Baire order of $H$ is $\omega_1$ and the $\sigma$-algebra
generated by $H$ is $P(\reals)$?
\topic{Involving $\omega_1$}
% 8.1
\prob (Jech-Prikry \cite{jp}) Is it consistent that
there exists a family $F\subset\omega^{\omega_1}$
of cardinality less than $2^{\omega_1}$, such that for every
$g\in\omega^{\omega_1}$ there exist $f\in F$ such that for every
$\alpha<\omega_1,\;\; g(\alpha)\omega_2$, and
there exists $F\subset[\omega_1]^{\omega_1}$ of cardinality
$\omega_2$ such that for every $A\subset\omega_1$ there exists
$B\in F$ such that $B\subseteq A$ or $B\cap A=\emptyset$?
% 8.4
\prob (Kunen) Is it consistent to have $2^{\omega_1}>\omega_2$ and
there exists $F\subset[\omega_1]^{\omega_1}$ of cardinality
$\omega_2$ such that for every uncountable $A\subset\omega_1$ there exists
$B\in F$ such that $B\subseteq A$?
% 8.5
\prob (Kunen) Is it consistent to have a uniform ultrafilter
on ${\omega_1}$ which is generated by fewer than $2^{\omega_1}$ sets?
% 8.6
\prob (Prikry \cite{m20}) Is it consistent there exists an
$\omega_1$ generated ideal J such that $P(\omega_1)=P(\omega)/J$?
% 8.7
\prob (Comer) If C and D are homeomorphic to $2^{\omega_1}$ then
is $C\cup D$? (Say if both are subsets of $2^{\omega_1}$.)
\answer{No, Bell \cite{bellem}.}
% 8.8
\prob (Nyikos) If $C\times D$ is homeomorphic to $2^{\omega_1}$ then
must either C or D be homeomorphic to $2^{\omega_1}$?
\answer{Yes, Bell \cite{bellem} see also
Bell \cite{bell-schep}, Schepin \cite{schep}.}
% 8.9
\prob (CH) Let n(X) be the cardinality of the smallest family of meager sets
which cover X. Can the cof(n($(2^{\omega_1})_{\delta}$))
($G_{\delta}$-topology) be $\omega$ or $\omega_1$?
% 8.10
\prob (Shelah \cite{shelah1976}) Is it consistent that every Aronszajn line L
contains a Countryman type?
\answer{Yes, PFA implies it, Moore \cite{moore}.}
% 8.11
\prob Does PFA imply that any two Aronszajn types contain uncountable
isomorphic subtypes?
\answer{Justin Moore points out that
the answer is: no, since no uncountable linear order
can embed into both a Countryman line and its converse.
However PFA implies that any two Aronszajn types contain uncountable
isomorphic or reverse isomorphic subtypes. This was
shown for Countryman types by Abraham and Shelah \cite{abrshel}
and follows
for Aronszajn lines from Moore's solution to problem 8.10.
}
% 8.12
\newprob
What is the exact consistency strength of PFA?
Schimmerling \cite{schimm} showed that PFA implies the
consistency of ZFC+$\exists$ a Woodin cardinal.
Shelah
building on work of Baumgartner (see \cite{shepfa}) showed that PFA is
consistent assuming the consistency of a supercompact cardinal.
% 8.13
\newprob
If the nonstationary ideal on $\omega_1$ is $\omega_2$-saturated,
then must CH fail? Woodin has recently shown that the answer
is yes if we also assume there is a measurable cardinal.
\topic{Set theoretic topology}
% 9.1
\prob Is it consistent to have no P-points or Q-points? A P-point
is an ultrafilter U on $\omega$ with the property that every
function $f:\omega\to\omega$ is either constant or finite-to-one
on an element of U.
A Q-point
is an ultrafilter U on $\omega$ with the property that every finite-to-one
function $f:\omega\to\omega$ is one-to-one
on an element of U.
Shelah \cite{shepfa} showed it is consistent there are no P-points
and Miller \cite{m2} showed that
it is consistent there are no Q-points. Roitman and Taylor
showed that if the continuum is $\leq \omega_2$, then there must
be a P-point or a Q-point.
% 9.2
\prob (M.E. Rudin) Is there always a small Dowker space?
\answer{Yes?, Balogh \cite{baldowk}, Kojman-Shelah \cite{kojshe}}
% 9.3
\prob (Charlie Mills) In infinite dimensional Hilbert space is a sphere
coverable by fewer than continuum other spheres?
% 9.4
\prob Is it consistent that $\omega^{\omega_1}$ is pseudonormal?
(Pseudonormal means disjoint closed sets can be separated if
at least one is countable.)
% 9.5
\prob (van Douwen) Is it consistent to have
$c(U(\omega_1))\mathfrak c$ many universally
measurable sets? (same question for restricted Baire property). There is a
model in which there are only continuum many universal measure zero
sets (see Miller \cite{m10}).
\answer{ See Larson, Neeman, and Shelah \cite{larson}. }
% 11.3
\prob (Galvin) Does every Sierpinski set have strong first category?
\answer{Bartoszynski-Judah \cite{bj} showed that it is consistently yes.
Yes, Pawlikowski \cite{paw}.}
% 11.4
\prob (Galvin, Carlson) Is the union of two strong first category
sets a set of strong first category?
\answer{Not necessarily, Bartoszynski-Shelah \cite{barshel}.}
% 11.5
\prob Does there exists a perfectly meager $X\subseteq\reals^n$ which is not
zero-dimensional?
Szpilrajn(Marczewski) proved that there is such a set assuming CH,
see Brown and Cox \cite{bc}. However is it consistent that there is none?
\answer{yes. Reclaw pointed out to me that:
A metric space of size less than continuum has to be zero dimensional
and it is consistent that all perfectly meager sets are of size less
than continuum, see Miller \cite{m10}.}
% 11.6
\prob (Kunen) Is it consistent that for every uncountable
$X\subseteq \reals$
there exists a measure zero set M such that $X+M$ has positive
outer measure? See Erdos-Kunen-Mauldin \cite{ekm}.
\answer{Yes, Carlson, this is true in the model
for the dual Borel conjecture \cite{dualborel},
(this was pointed out to me by Brendle and Reclaw.)}
% 11.7
\prob (Sierpi\'{n}ski \cite{sier}) A set of reals X is a J-set iff for
uncountable
$Y\subseteq X$ there exist a perfect $P\subseteq X$ such that $P\cap Y$
is uncountable. If we assume CH, then a set is a J-set iff it is
$\sigma$-compact. Is it consistent with not CH that
J-set $ = \sigma$-compact?
% 11.8
\prob (Fremlin-Miller \cite{m22})
Is there always an uncountable subset of the reals which is hereditary
with respect to property M?
\answer{See \cite{m34} for some related results.}
% 11.9
\prob Consider the three non c.c.c ideals: $(s)_0$-sets, Ramsey null sets,
and $\sigma$-compact sets.
What can one say about the properties of add, cov, non,
and cof? ( add = additivity of ideal, cov = smallest cardinality of a
cover of reals by subset of ideal, non = smallest cardinality of a
set of reals not in ideal,
cof = cofinality = smallest cardinality of a family of sets in ideal
which has the property
that every set in ideal is covered by some element of the family.)
% 11.10
\prob Consider the notion of Laver null sets. This is defined analogously
to Ramsey null sets, but use Laver forcing instead of Mathias forcing.
The analogue of Galvin-Prikry Theorem is true here. What other results
also go thru? What ideals arise from other notions of forcing?
What about Silver forcing?
What notions of forcing arise from infinite combinatorial theorems?
(For example, Carlson's infinite version of the Hales-Jewett theorem
\cite{carl}.)
\answer{For some work in this direction, see L\"{o}we \cite{low} and
Brendle and L\"{o}we \cite{brelow}.}
% 11.11
\prob (Judah-Shelah \cite{js}) Is the Borel conjecture plus the existence of
a Q-set consistent?
% 11.12
\prob (Daniel, Gruenhage). Given a set of reals $X$ and ordinal
$\alpha$ let $G_{\alpha}(X)$ be the game of length $\alpha$ played
by two players: point picker and open. At each play of the game
point picker picks a real and open responds with an open set
including the real. Point picker wins a run of the game if at the
end the open sets chosen cover $X$. The order of $X$ is the least
ordinal for which point picker has a winning strategy. What orders
are possible? Daniel and Gruenhage have examples of order
$\omega n$ assuming CH.
\answer{All countable limit ordinals are possible, Baldwin \cite{baldwin}.}
% 11.13
\newprob (Komjath, see Steprans \cite{stepkom},\cite{stepr})
Suppose every set of reals of size $\omega_1$ has measure zero.
Then does every $\omega_1$ union of lines have planar measure zero?
(Dually) Suppose the real line is union of $\omega_1$ measure zero sets.
Then does there exists $\omega_1$ measure zero subsets of the plane
such that every line is contained in one of them?
\par The answer is known if line is changed to graph of continuous function,
see Bartoszynski, Roslanowski, Shelah \cite{brs}.
There is also the analagous question for category.
\answer{Shelah-Steprans \cite{shelahstep},
see also Steprans \cite{step2005}.}
% 11.14
\newprob (Zhou email 3-93)
Does every set of size $\omega_1$ is a Q-set imply that
${\mathfrak p} > \omega_1$.
For $\gamma$-sets it is true.
\answer{No, Brendle points out this follows from
results in Dow \cite{dow33}. We don't know if
``every set of size $\omega_1$ is a Q-set'' implies ``the real
line cannot be covered by $\omega_1$ meager sets''.
It is consistent to have a Q-set plus ``the real
line can be covered by $\omega_1$ meager sets'', see
Judah-Shelah \cite{js}.}
% 11,15
\newprob
(M.Laczkovich, email April 1996)
Assuming CH, there is a nonmeasurable subset of the
reals that differs from each of its translates
in a set of measure zero (Sierpinski). Can
such a set can be given in ZFC?
\answer{No, see Laczkovich, \cite{laksier}.}
% 11.16
\newprob (A.Marcone, email May 1997)
Is it consistent that every $\lambda$-set (space) is a $Q$-set (space)?
\topic{Quasiorder theory}
% 12.1
\prob Is there a Borel version of Fraisse's conjecture?
Are the Borel linear orderings well-quasiordered under embedding?
\answer{Yes, Louveau and St-Raymond \cite{louv} assuming large parts of AD.
See also Louveau \cite{louv2}.}
% 12.2
\prob (Laver) Is it consistent that the set of Aronszajn trees is
well-quasi-ordered under embeddability? See Laver \cite{lav}
and Corominas \cite{coromin}.
\answer{No, Todorcevic \cite{tod98}. Under PFA the Aronszajn lines
are wqo, Martinez-Ranero \cite{martinez}.}
% 12.3
\prob (Kunen) Is the set of all better-quasi-ordered binary relations
on $\omega$ a proper $\Pi^1_2$ set?
\answer{Yes, Marcone \cite{marco}.}
% 12.4
\prob Suppose $(Q,\leq)$ is a recursive quasi-order. Is it true that
Q is BQO iff $Q^{<\omega_1^{ck}}$ is WQO?
% 12.5
\prob (Kunen) Suppose $(Q,\leq)$ is a recursive well quasi-order. Does
$Q^{\omega}/\equiv$ have a recursive presentation?
(It is countable, see Laver \cite{FC} Theorem 4.11 for wqo.)
% 12.6
\prob Suppose every set is Ramsey and $f:\infsets\to\mbox{ORD}$.
Then does there exist $X\in\infsets$ such that the image
of $[X]^\omega$ under $f$ is
countable? See Louveau-Simpson \cite{simp} and
Aniszcyk-Frankiewicz-Plewik \cite{afp}.
% 12.7
\prob Is finite graphs under homeomorphic embedding WQO?
% 12.8
\prob Is the witness lemma true for LIN(Q) or TREE(Q)?
% 12.9
\prob Is there an $\omega_1$-descending sequence of countable posets
(under embedding) each of which is the union of two chains?
(Kunen, Miller: There is an $\omega_1$-descending sequence of
countable posets. Kunen: There is an infinite antichain of finite
posets each of which is the union of two chains.
The first result appears in the second edition of Fra\"iss\'e's book
\cite{fraisse}.)
% 12.10
\prob Is there a parameterized version of Carlson's theorem?
See Carlson \cite{carl} and Pawlikowski \cite{pawpara}.
\topic{not AC}
% 13.1
\prob (Bell) Let C stand for:
\par (C) For every family of nonempty sets
there exists a function assigning to each set in the family
a compact Hausdorff topology.
\par Is (C) equivalent to AC? If not, what principles of
choice is (C) equivalent to?
\par\medskip Motivation: PIT (Prime Ideal Theorem)
is equivalent to every Tychonov product of compact Hausdorff spaces
is compact. Hence AC is equivalent to C+PIT.
In an earlier version of these problems I had mistakenly
written ``Does ZF prove C?''. However,
it is known that PIT does not imply AC (see Jech \cite{jechac}), hence
C fails in any model of ZF + PIT + notAC.
See Howard and Tachtsis \cite{howard}.
% 13.2
\prob (Dow 88) Does
Stone's theorem on metric spaces
(every metric space is paracompact) require AC?
It is known that ZF implies that $\omega_1$ with the
order topology is not metrizable.
\answer{Yes, Good, Tree, and Watson \cite{good}.}
% 13.3
\newprob (Morillon \cite{morillon}) (In ZF) does every
compact Hausdorff space which is countable have an isolated point?
% 13.4
\newprob
(M.Bell, email April 96) Is it consistent to have the prime ideal theorem
plus there does not exist an $\omega_1$ descending sequence of distinct sets
mod finite, i.e.,
$\langle A_\alpha\in[\omega]^\omega:\;\alpha<\omega_1\rangle$ with
$A_\alpha\subseteq^*A_\beta$ for $\beta<\alpha$? Bell notes that in
such a model of set theory, we would have that $\beta\omega\setminus \omega$
would exist in all its glory, but hardly anything of the standard stuff
about it could be proved.
\topic{Recursion theory}
% 14.1
\prob Does there exist a non-trivial automorphism of the Turing degrees?
(Re degrees?)
A Yes answer was announced by Cooper in 1995
and published in Cooper \cite{cooper},
but nobody else seems to believe it is a correct proof.
% 14.2
\prob (Jockusch) Does there exists a DNR of minimal Turing degree?
(DNR means diagonally non recursive: $f\in\bsp$ and
for all $e\in\omega$,
$\;\;f(e)\not=\{e\}(e)$.)
\answer{Kumabe and Lewis \cite{kumabe}}
\topic{Miscelleneous}
% 15.1
\prob (Sierpi\'{n}ski) Is there a Borel subset of the plane which
meets every line in exactly two points?
(Mauldin) Must such a set be zero dimensional?
Davies has shown such a set cannot be $\Sigma^0_2$ and Mauldin
has shown such a set must be disconnected. Miller \cite{m24} showed
that if V=L then there does exist a $\Pi^1_1$ subset of the plane
which meets every line in exactly two points. Kulesza
\cite{kul} showed that any two point set must be zero dimensional.
Mauldin \cite{maul2pt}, Dijkstra, Kunen, van Mill, \cite{dij}
\cite{dkm} \cite{dm}, have some recent work.
% 15.2
\prob (Cichon) Is it consistent to have that the real line is the
disjoint union of $\omega_2$ meager sets such that every meager
set is contained in a countable union of them?
\answer{No, Brendle \cite{brendcic}.}
% 15.3
\prob (Juhasz) Does club imply there exist a Souslin line?
\answer{Maybe No, Dzamonja-Shelah \cite{dzsh}.
There seems to be a mistake in their proof. I do not know
the current status of this problem.}
% 15.4
\prob (Ulam \cite{ulam}) Does there exist a set D dense in the plane
such that the distance between any two points of D is rational?
% 15.5
\prob (Miller \cite{m10})
Suppose the continuum is greater than $\omega_2$, then does there
exists a set of reals of cardinality the continuum which cannot be
mapped continuously onto the unit interval?
% 15.6
\prob Is it consistent that there exists $x\in\csp$ such
that $V=L[x]\neq L$ and a continuous onto function
$f:L\cap\csp\to V\cap\csp$?
% 15.7
\prob (Price \cite{price}) Is it consistent there is no Cech function?
\answer{There is a Cech function in ZFC, see Galvin and Simon
\cite{galvinsimon}.}
% 15.8
\prob (Kunen) Does the consistency of an elementary embedding of $M$ into $V$
imply the consistency of a measurable cardinal?
\answer{No, Vickers and Welch \cite{welch} show a Ramsey is enough.
See also Suzuki \cite{suzuki}.}
% 15.9
\prob (Erd\"{o}s) Without CH can you partition the plane into countably many
pieces so that no piece contains an isoceles triangle?
See Kunen \cite{kuntri}.
\answer{Yes, Schmerl \cite{schm}.}
% 15.10
\prob Is there a Borel version of Hall's marriage theorem? As for
example, the Borel-Dilworth Theorem \cite{BDT}.
\answer{Kechris pointed out to me that the answer to this is no. It follows
from a result of Laczkovich \cite{lacz}. In Laczkovich's example $R$
a matching is required to be both one-to-one and onto. To get a counterexample
to a Borel version of Hall's theorem take $R$ and its reverse $R^*$
in $I\times I$. Both satisfy the
hypothesis of Hall's Thm. But if there are Borel $f$, $f^*$ 1-1 with
graphs in $R$, $R^*$ respectively,
then we can find Borel one-one onto $g$ with graph($g$) a
subset graph($f)\cup$ graph($f^{*-1})$.
See also Andrew Marks \cite{marks}.
}
% 15.11
\prob (Davies \cite{davies}) Assuming CH for every
$f:\reals^2\to \reals$ there exists $g_n,h_n:\reals\to \reals$
such that
$$f(x,y)=\Sigma_{n\in\omega}g_n(x)h_n(y)$$
Does this imply CH?
\answer{No and its also consistently false Shelah \cite{shelahdavies}.}
% 15.12
\prob (Mauldin) CH implies that for every $n\geq 3$ there exists
a 1-1 onto function $f:\reals^{n}\to \reals^{n}$ which maps
each circle onto
a curve which is the union of countably many line segments. Is CH
necessary?
% 15.13
\prob (Kunen) Can there be a Souslin tree $T\subseteq 2^{\kappa}$ such
that for all $\alpha<\kappa$ the $T_{\alpha}$ contains all except at
most one of the $\alpha$ branches thru $T_{<\alpha}$.
Here $\kappa$ is the first Mahlo or weakly Mahlo.
\answer{Yes, Shelah \cite{shelsous}.}
% 15.14
\prob (Baumgartner \cite{baum}) Is it consistent that any two
$\omega_2$ dense sets
of reals are order isomorphic?
% 15.15
\prob (S. Kalikow \cite{kal}) For any set $X$ define for $x,y\in X^{\omega}$,
$\;\;x=^*y$ iff for all but finitely many $n\in\omega$, $\;\;
x(n)=y(n)$. $X$ has the discrete topology and
$X^{\omega}$ the product topology.
Is it consistent that there exists a map
$f:\omega_2^{\omega}\to \csp$ which is continuous
and for every $x,y\in {\omega_2}^{\omega}$, $\;\; x=^*y$ iff $f(x)=^*f(y)$.
(Kalikow: yes for $\omega_1$ in place of $\omega_2$.)
\answer{Yes, Shelah \cite{shel4}.}
% 15.16
\newprob (unknown 1-92) According to Erdos,
Sylvestor proved that given finitely many
points $F$ in the plane not all collinear, there exists a line
$L$ which meets $F$ in exactly two points. $F={\mathbb Z}\times {\mathbb Z}$
is an obvious infinite counterexample. Does there exists
a counterexample which is a convergent sequence? countable compact
set?
\answer{Marton Elekes (email Dec 2005) informs me that there is
such a sequence. He does not intend to publish since even
it is not known, it is far too simple.
Using projective geometry and duality, he came up with
the example:
$\{(0,0),\; (\frac1{3n+1},0),\; (0,\frac1{3n+1}),\;
(\frac1{3n+2},\frac1{3n+2})\;\;:\;
n\in{\mathbb Z}\}$.
To check that this works note that the line containing $(0,\frac1{3n+1})$ and
$(\frac1{3k+1},0)$ is
$(3n+1)y+(3k+1)x=1$ and meets $x=y$ at $(a,a)$ with $a=\frac1{3n+3k+2}$.
Conversely given $(\frac1{3k+1},0)$ and $(a,a)$ with $a=\frac1{3m+2}$
put $n=m-k$.
}
% 15.17
\newprob Given that $2^{\aleph_n}=\aleph_{n+1}$ for each
$n<\omega$ what can we say about $2^{\aleph_\omega}$?
\par\noindent
Shelah has shown that if $\aleph_\omega$ is
a strong limit cardinal, then $2^{\aleph_\omega}<\aleph_{(2^{\aleph_0})^+}$
and $2^{\aleph_\omega}<\aleph_{\aleph_4}$.
See Shelah \cite{shelpcf} or Jech \cite{jech} or
Burke-Magidor \cite{burmag}. On the other
hand Gitik-Magidor \cite{gitmag} have shown that is consistent
relative to the existence of strong cardinals that
$2^{\aleph_\omega}=\aleph_{\omega+\zeta+1}$ for any
$\zeta<\omega_1$. What about the gap?
See Jech-Shelah \cite{jecshe}.
Also, many variations
on this questions can be given. For example,
Shelah has shown that relative to a supercompact
it is consistent that for the least uncountable $\kappa$ with
$\aleph_\kappa=\kappa$ that the GCH holds up to $\kappa$ but
$2^\kappa$ can be arbitrarily large. What about singular cardinals
in between? What are the exact consistency strengths of
these statements? Magidor-Gitik \cite{gitmag}
have gotten Shelah's result from a weaker assumption.
Gitik \cite{git} building on work of
Mitchell \cite{mit} has shown for example that the
existence of a measurable $\kappa$ with $o(\kappa)=\kappa^{++}$
is equiconsistent with the failure of the singular cardinal
hypothesis. For more on this see Cummings \cite{cum}.
% 15.18
\newprob (Dougherty-Kechris \cite{dk})
Is Turing equivalence is universal for countable Borel
equivalence relations, i.e., for every countable Borel
equivalence relation $(X,E)$ does there exists a 1-1 Borel
map $f:X\to 2^\omega$ such that for all $u,v\in X$
$$uEv \mbox{ iff } f(u)\equiv_T f(v).$$
The countable Borel equivalence relations are those in which
every equivalence class is countable.
See Kechris \cite{kecmsri}, \cite{kectapp}, and
Harrington, Kechris, and Louveau \cite{hkl} for some background here.
% 15.19
\newprob (Kechris, Solecki, and Todorcevic) Is it
possible to have a Borel graph with coloring number
$2$ but Borel coloring number $4$? They have examples
for $n$ and $n+1$.
\answer{Yes, Laczkovich, see the appendix of \cite{stk}.}
% 15.20
\newprob
(M.Bell, letter to M.E.Rudin, April 1996)
Can there exist a cardinal $\kappa >{\mathfrak c}$ for
which there exists
$\kappa^+$ subsets of $\kappa$ each of cardinality $\kappa$ and with
pairwise intersection finite?
\bigskip\bigskip
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D.H.Fremlin, Problems to add to the gaiety of nations,
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Victoria Delfino Problems, Cabal Seminar, Lecture notes in
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\ref
{\bf Analytic Sets}, edited by C.A.Roger, J.E.Jaynes, Academic Press, 1980.
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\begin{flushright}
Arnold W. Miller\\
University of Wisconsin\\
Department of Mathematics, Van Vleck Hall\\
480 Lincoln Drive\\
Madison, WI 53706\\
miller@math.wisc.edu\\
http://www.math.wisc.edu/$\sim$miller\\
\end{flushright}
\end{document}