\documentclass[12pt,leqno]{article} \textwidth=30cc \baselineskip=16pt \newcommand{\UpdateDate}{Aug 25, 03} \date{} \title{A big symmetric planar set with small category projections} \pagestyle{myheadings} \markboth{K.~Ciesielski, T.~Natkaniec \ \ \ \ \ \UpdateDate }{A big symmetric set with small category projections \ \UpdateDate } \newcommand{\reap}{{\mathfrak r}} \newcommand{\ultraN}{{\mathfrak u}} \newcommand{\spleat}{{\mathfrak s}} \newcommand{\madN}{{\mathfrak a}} \newcommand{\indN}{{\mathfrak i}} \newcommand{\mathP}{{\mathbb P}} \renewcommand{\H}{{\cal H}} \newcommand{\U}{{\cal U}} \newcommand{\V}{{\cal V}} \newcommand{\E}{{\cal E}} \newcommand{\forces}{\Vdash} \newcommand{\psmCGame}{{\rm CPA$_{\rm cube}^{\rm game}$}} \newcommand{\psmC}{\mbox{{\rm CPA$_{\rm cube}$}}} \def\range{\mathop{\rm range}} \author{ Krzysztof Ciesielski% \thanks{The work of the first author was partially supported by NATO Grant PST.CLG.977652 and 2002/03 West Virginia University Senate Research Grant. }\\ {\footnotesize Department of Mathematics,} {\footnotesize West Virginia University,} \\ {\footnotesize Morgantown, WV 26506-6310, USA}\\ {\footnotesize e-mail: K\_Cies@math.wvu.edu}; %\\ {\footnotesize web page: {\tt http://www.math.wvu.edu/\~{}kcies}} \and Tomasz Natkaniec% \thanks{Partially supported by Gda{\'n}sk University Research Grant No. BW 5100-5-02331-2. }\\ {\footnotesize Department of Mathematics,} {\footnotesize Gda\'{n}sk University,} \\ {\footnotesize Wita Stwosza 57, 80--952 Gda\'{n}sk, Poland} \\ {\footnotesize e-mail: mattn@math.univ.gda.pl;}} \usepackage{amsmath} \usepackage{amssymb} \usepackage{amsthm} \newtheorem{thm}{Theorem} \newtheorem{cor}[thm]{Corollary} \newtheorem{prop}[thm]{Proposition} \newtheorem{lem}{Lemma} \newtheorem{re}{Remark} \newtheorem{pr}{Problem} \newtheorem{ex}{Example} \newtheorem{df}{Definition} \newtheorem{conj}{Conjecture} \newtheorem{Fact}[thm]{Fact} \newcommand{\fact}[2]{\begin{Fact}\label{#1}{\sl #2}\end{Fact}} \def\dom{{\rm dom}} \newcommand{\la}{\langle} \newcommand{\ra}{\rangle} \def\continuum{{\mathfrak c}} \newcommand{\mathR}{{\mathbb R}} \newcommand{\real}{{\mathbb R}} \newcommand{\mathQ}{{\mathbb Q}} \newcommand{\RRR}{\mathR^{\mathR}} \newcommand{\co}{{\mathfrak c}} \newcommand{\aaa}{{\alpha}} \newcommand{\bbb}{{\beta}} \renewcommand{\ggg}{{\gamma}} \newcommand{\ddd}{{\delta}} \newcommand{\fff}{{\varphi}} \newcommand{\kkk}{{\kappa}} \renewcommand{\lll}{{\lambda}} \newcommand{\ooo}{{\omega}} \newcommand{\eee}{{\varepsilon}} \newcommand{\card}{\mbox{\/\rm card}\,} \newcommand{\add}{\mbox{\/\rm add}\,} \newcommand{\cov}{\mbox{\/\rm cov}\,} \newcommand{\non}{\mbox{\/\rm non}\,} \newcommand{\osc}{\mbox{\/\rm osc}\,} \newcommand{\cl}{\mbox{\/\rm cl}\,} \newcommand{\dist}{\mbox{\/\rm dist}\,} \newcommand{\F}{{\cal F}} \newcommand{\G}{{\cal G}} \newcommand{\I}{{\cal I}} \newcommand{\B}{{\cal B}} \newcommand{\M}{{\cal M}} \newcommand{\K}{{\cal K}} \newcommand{\C}{{\cal C}} \newcommand{\A}{{\cal A}} \newcommand{\Q}{{\cal Q}} \newcommand{\J}{{\cal J}} \renewcommand{\L}{{\cal L}} \renewcommand{\P}{{\cal P}} \newcommand{\N}{{\cal N}} \renewcommand{\int}{\mbox{\/\rm int}\,} \newcommand{\LIM}{{\rm LIM}_{\ooo_1}\,} \newcommand{\pf}{{\noindent\sc Proof. }} \newcommand{\id}{{\rm id}} \def\Cantor{{\mathfrak C}} \newcommand{\perf}{{\rm Perf}} \newcommand{\Fcube}{{\cal F}_{\rm cube}} \newcommand{\Ccube}{{\cal C}_{\rm cube}} \begin{document} { \renewcommand{\thefootnote}{} \footnotetext{AMS Subject Classification (1991): Primary {\bf 03E05}; Secondary: 03E50; 03E75.} } {\renewcommand{\thefootnote}{} \footnotetext{Key words: $f$-category projections; transfinite induction; nowhere meager sets. Covering Property Axiom CPA; $\omega_1$-oracle.} } \maketitle \begin{abstract} We show that under appropriate set theoretic assumptions (which follow from Martin's axiom and the continuum hypothesis) there exists a nowhere meager set $A\subset\mathR$ such that \begin{itemize} \item[(i)] for each continuous nowhere constant function $f\colon\mathR\to\mathR$ the set $\{c\in\mathR\colon \pi[(f+c)\cap (A\times A)]\mbox{ is not meager}\}$ is meager, and \item[(ii)] the set $\{c\in\mathR\colon (f+c)\cap (A\times A)=\emptyset\}$ is nowhere meager for each continuous function $f\colon\mathR\to\mathR$. \end{itemize} The existence of such a set follows also from the principle CPA, which holds in the iterated perfect set model. We also prove that the existence of a set $A$ as in (i) cannot be proved in ZFC alone even when we restrict our attention to homeomorphisms of $\real$. On the other hand, for the class of real analytic functions a Bernstein set $A$ satisfying (ii) exists in ZFC. \end{abstract} \section{The results} Let $\M$ denote the class of all meager subsets of $\mathR$ and let $\pi\colon\mathR^2\to\mathR$ be the projection into the first coordinate. For a function $f\colon\mathR\to\mathR$ and a set $E\subset\mathR^2$ the {\em $f$-category projection}\/ of $E$ is the set $\{c\in\mathR\colon \pi[(f+c)\cap E]\notin\M\}$. (See \cite{CG}.) In papers \cite{RD}, \cite{TN}, and \cite{TN1} the authors considered the second category sets $A\subset\mathR$ such that an $f$-category projection of $A\times A$ have an empty interior for every linear function $f(x)=ax+b$. Inspired by these results Bartoszy{\'n}ski and Halbeisen~\cite{BH} recently constructed a second category set $A\subset\mathR$ with even stronger property: such that for each polynomial $p$ which is neither constant nor the identity function the set $p\cap(A\times A)$ is finite. In particular, if $p$ is a non-constant polynomial then the $p$-category projection of $A\times A$ has an empty interior. These results lead to the following, more general question: for which classes $\F$ of continuous functions does there exist a ``big'' set $A\subset\real$ such that the $f$-category projection is ``small'' for every $f\in\F$? In particular, what happens for the classes $\A$ of real analytic functions, $\C_0$ of all nowhere constant functions from $\mathR$ to $\mathR$, and for the entire class $\C$ of all continuous functions from $\mathR$ to $\mathR$? In this note we answer these questions. We use standard terminology as in \cite{CiBook}. We consider only real-valued functions of one variable, unless otherwise specified. No distinction is made between a function and its graph. For functions $f,g$ put $[f=g]=\{ x\in\mathR\colon\; f(x)=g(x)\}$. A set $A\subset\mathR$ is nowhere meager if $A\cap I\not\in\M$ for each non-degenerate interval $I$. Symbol $B(x,r)$ denotes the open ball centered at $x$ and with the radius $r$ and $\id$ stands for the identity function. The main general theorem in a positive direction is the following result. \begin{thm}\label{tmain} Let $\J\supset[\real]^{\leq\omega}$ be a translation invariant ideal on $\real$ and $B_0$ be a family of Borel sets containing all Borel non-meager sets such that \begin{equation} \label{eqMain1} \mbox{$B\setminus\bigcup\G\neq\emptyset$ for every $B\in\B_0$ and $\G\in[\J]^{<\co}$.} \end{equation} If $\F\subset\C$ is such that for every $f\in\F$ \begin{equation}\label{eqMain2} \mbox{the set $L_f=\{y\in\mathR\colon f^{-1}(y)\notin\J\}$ belongs to $\J$} \end{equation} then there exists an $A\subset\mathR$ intersecting every $B\in\B_0$ such that for every $f\in\F$ \begin{itemize} \item[{\rm (a)}] there exists a $Z\in[\real]^{<\co}$ such that $f\cap(A\times A)\subset(A\times Z)\cup\id$. \end{itemize} Moreover, if\/ {\rm (\ref{eqMain2})} holds for every $f\in(\pm\F)\cup(\id-\F)$ then we can also have the following property for every $f\in\F$ \begin{itemize} \item[{\rm (b)}] the set $\{ c\in\mathR\colon (f+c)\cap (A\times A)=\emptyset\}$ intersects every $B\in\B_0$. \end{itemize} \end{thm} We leave the proof of this theorem to the next section. In the reminder of this section we like to discuss several of its corollaries. In particular, applying Theorem~\ref{tmain} to the ideal $[\real]^{\leq\omega}$ of countable sets and the family $\B_0$ of all uncountable Borel sets we get the following result. \begin{cor}\label{cor1} Let $\F\subset\C$ be such that the set $L_f=\{y\in\mathR\colon |f^{-1}(y)|>\omega\}$ is empty for every $f\in\F$. Then there exists a Bernstein set $A\subset\mathR$ such that for every $f\in\F$ \begin{itemize} \item[{\rm (a)}] there exists a $Z\in[\real]^{<\co}$ such that $f\cap(A\times A)\subset(A\times Z)\cup\id$. \end{itemize} In addition, if $L_f=\emptyset$ for every $f\in(\pm\F)\cup(\id-\F)$ then we can also have \begin{itemize} \item[{\rm (b)}] $\{c\in\mathR\colon (f+c)\cap (A\times A)=\emptyset\}$ contains a Bernstein set for every $f\in\F$. \end{itemize} In particular, \begin{itemize} \item there exists a set $A$ satisfying\/ {\rm (a)} for every countable-to-one function $f\in\C$; \item there exists a set $A$ satisfying\/ {\rm (a)} and\/ {\rm (b)} for every analytic function $f\in\A$; moreover, if $f\neq\id$ is not constant then $|f\cap(A\times A)|<\co$. \end{itemize} \end{cor} \pf Clearly the ideal $\J=[\real]^{\leq\omega}$ and the family $\B_0$ of all uncountable Borel sets satisfy (\ref{eqMain1}) from Theorem~\ref{tmain}. Since we assume that (\ref{eqMain2}) is also satisfied, the main part of the corollary follows Theorem~\ref{tmain}. To see the additional part for analytic functions first notice that $\F=\A$ satisfies the assumption for the part (b), since the set $L_f=\emptyset$ is empty for every non-constant $f\in\A$. In particular, if $f\neq\id$ then, by condition (b), the set $f\cap(A\times A)$ is contained in $(f-\id)^{-1}(0)\cup\bigcup_{z\in Z}f^{-1}(z)$, which is a union of less than $\co$-many countable sets. \qed In the proof of the next corollary we need the following simple fact. \begin{equation}\label{eqFact} \mbox{If $f\in\C$ then the set $\{ c\colon\; [f+c=\id]\not\in\M\}$ is countable.} \end{equation} To see it notice that for each $c\in\mathR$ the set $[f+c=\id]$ is closed. Therefore, if $[f+c=\id]\not\in\M$, there exists a non-empty open interval $I_c$ with $f(x)=x-c$ for each $x\in I_c$. It is easy to observe that $I_c\cap I_d=\emptyset$ if $c\neq d$. \begin{cor}\label{cor1b} There exists a Bernstein set $A\subset\real$ such that for every homeomorphism $f\colon\real\to\real$ for all but countably many $c\in \real$ the set $\pi[(f+c)\cap (A\times A)]$ is a union of a meager set and a set of cardinality less than $\co$. In particular, if $[\real]^{<\co}\subset\M$ then the set \[ \{c\in\mathR\colon \pi[(f+c)\cap (A\times A)]\notin\M\} \] is countable (so meager) for every homeomorphism $f\colon\real\to\real$. \end{cor} \pf Let $A$ be as in Corollary~\ref{cor1}. So for every homeomorphism $f\colon\real\to\real$ and $c\in\real$ there exists a $Z\in[\real]^{<\co}$ such that $(f+c)\cap(A\times A)\subset(A\times Z)\cup\id$. Notice that $\pi[(f+c)\cap(A\times A)]\subset[f+c=\id]\cup\pi[(f+c)\cap(A\times Z)]$ and $\pi[(f+c)\cap(A\times Z)]\subset f^{-1}(Z-c)\in[\real]^{<\co}$. So there exists a $D\in[\real]^{<\co}$ such that $\pi[(f+c)\cap(A\times A)]\subset[f+c=\id]\cup D$. The conclusion follows from~(\ref{eqFact}). \qed Notice that the set theoretical assumption in Corollary~\ref{cor1b} is essential. \begin{thm}\label{thmCons1} It is relatively consistent with the set theory ZFC that for every nowhere meager set $A\subset\mathR$ there exists a homeomorphism $f\colon\real\to\real$ such that $\{c\in\real\colon \pi[(f+c)\cap(A\times A)]\notin\M\}$ is nowhere meager. \end{thm} \pf This follows immediately from Theorem~\ref{thConsOracle} and Proposition~\ref{propA} which will be proven in Section~\ref{sec:oracle}. \qed Applying Theorem~\ref{tmain} to the ideal $\M$ and the family $\B_0$ of non-meager Borel sets we get also the following result. (Recall that the set theoretic assumption about covering by meager sets follows form the Martin's axiom MA and from the continuum hypothesis CH.) \begin{cor}\label{cor2} Assume that less than continuum many meager sets do not cover $\real$. Then there exists a nowhere meager set $A\subset\mathR$ such that for every $f\in\C$ the set $\{c\in\mathR\colon (f+c)\cap (A\times A)=\emptyset\}$ is nowhere meager in $\mathR$. In particular, the $f$-category projection of $A\times A$ has an empty interior. \end{cor} \pf The set theoretical assumption assures that the ideal $\J=\M$, the family $\B_0$ of non-meager Borel sets, satisfy assumptions (\ref{eqMain1}) from Theorem~\ref{tmain}. Since, for $\J=\M$, (\ref{eqMain2}) holds for every $f\in\C$, there is a set $A$ satisfying (b) from Theorem~\ref{tmain}. Clearly, it has the desired properties. \qed In case of the class $\C_0$ of nowhere constant functions we can have yet another corollary. In its proof we will use the following simple fact. \begin{equation}\label{PreimM} f^{-1}(M)\in\M \mbox{ for every $f\in\C_0$ and $M\in\M$}. \end{equation} Indeed, if sets $F_n$ are closed and nowhere dense in $\real$ such that $M\subset \bigcup_{n<\omega}F_n$ then $f^{-1}(M)\subset\bigcup_{n<\omega}f^{-1}(F_n)$. It is just enough to notice that $f^{-1}(F_n)$ is closed and nowhere dense for every $f\in\C_0$. \begin{cor}\label{cor3} Assume that less than continuum many meager sets do not cover $\real$ and that $[\real]^{<\co}\subset\M$. Then there exists a nowhere meager set $A\subset\mathR$ such that for every $f\in\C_0$ the set $\{c\in\mathR\colon \pi[(f+c)\cap (A\times A)]\notin\M\}$ is countable. \end{cor} \pf Let $f\in\C_0$ and $c\in\real$. By part (a) of Theorem~\ref{tmain} there exists a $Z\in[\real]^{<\co}$ such that $\pi[(f+c)\cap(A\times A)]\subset[f+c=\id]\cup\pi[(f+c)\cap(A\times Z)]$. But, by (\ref{PreimM}), $\pi[(f+c)\cap(A\times Z)]\subset f^{-1}(Z-c)\in\M$ since $Z-c\in[\real]^{<\co}\subset\M$. Thus, $\pi[(f+c)\cap(A\times A)]\in\M$ as long as $[f+c=\id]\in\M$. So, the result follows immediately from (\ref{eqFact}). \qed We believe that the conclusion of Corollary~\ref{cor3} cannot be proved in ZFC, what we state as a following conjecture. (See also the last section of the paper for some comments on it.) \begin{conj}\label{conj1} It is relatively consistent with ZFC that for every nowhere meager set $A\subset\mathR$ there is an $f\in\C_0$ such that $\pi[(f+c)\cap(A\times A)]\notin\M$ for every $c\in\real$. \end{conj} It worth to note that a set $A$ as in Corollaries~\ref{cor2} and~\ref{cor3} can be also constructed under the Covering Property Axiom CPA, which extracts the essence of the iterated perfect set model. (See \cite{CP83,CMP90,CPAbook}.) This is of interest, since under CPA the set theoretical assumptions of each of these corollaries are false: CPA implies that $\co=\omega_2$ and that $\mathR$ can be covered by $\omega_1$ meager sets. In fact, in the theorem we will use only a simpler version of CPA known as \psmCGame. \begin{thm}\label{thmCPA} Assume that \psmCGame\ holds. Then there exists a nowhere meager set $A\subset\mathR$ of cardinality $\omega_1<\continuum$ such that for every $f\in\C$ there exists a countable set $Z\subset A$ such that \[ f\cap(A\times A)\subset(A\times Z)\cup\id. \] In particular, for every $f\in\C_0$ the set $\{c\in\mathR\colon \pi[(f+c)\cap (A\times A)]\notin\M\}$ is countable and the set $\{c\in\real\colon (f+c)\cap(A\times A)=\emptyset\}$ is a complement of a set of cardinality $\omega_1<\continuum$, so it contains a Bernstein set. \end{thm} \pf The main part of the theorem will be proved in Section~\ref{sec:CPA}. To prove the additional part of the theorem fix an $f\in\C_0$. The proof that the set $\{c\in\mathR\colon \pi[(f+c)\cap (A\times A)]\notin\M\}$ is countable is exactly the same as for Corollary~\ref{cor3}. To see that the complement $D$ of $\{c\in\real\colon (f+c)\cap(A\times A)=\emptyset\}$ has cardinality $\omega_1$ notice that if $(f+c)\cap(A\times A)\neq\emptyset$ then there are $x,y\in A$ such that $f(x)+c=y$, that is, $c=y-f(x)\in A - f[A]$. So, $D$ is equal to the set $A - f[A]$ and has cardinality~$\omega_1$. \qed It is also good to notice that the set $A$ in the Corollaries~\ref{cor2} and~\ref{cor3} cannot have the Baire property. \begin{prop}\label{prop2AAA} Suppose that $f\in\C_0$ and that $A\subset\mathR$ is a non-meager set having the Baire property. Then the $f$-category projection of $A\times A$ has a non-void interior. \end{prop} \pf Let $G$ be a non-empty open set with $M=G\setminus A\in\M$. Fix $x_0\in G\cap A$ and $c_0\in\mathR$ with $f(x_0)+c_0\in G\cap A$, an $\varepsilon>0$ such that $B(f(x_0)+c_0,2\varepsilon)\subset G$, and a $\delta>0$ such that $f[B(x_0,\delta)]\subset B(f(x_0),\varepsilon)$. Let $c\in B(c_0,\varepsilon)$. Since $A_0=A\cap B(x_0,\delta)\not\in\M$ and $f+c\in\C_0$, condition (\ref{PreimM}) implies $(f+c)(A_0)\not\in\M$. Since $(f+c)(A_0)\subset G$ and $(f+c)(A_0)\cap A\not\in\M$ point $c$ belongs to the $f$-category projection of $A\times A$. \qed We will finish this section with following result of Bartoszynski and Halbeissen, which was one of the departing points for this note. \begin{thm}\label{prop2}{\rm \cite{BH}} There exists a set $A\subset\mathR$ intersecting every perfect set such that for each non-constant polynomial $p\neq\id$ the set $p\cap(A\times A)$ is finite. \end{thm} Note that Corollary~\ref{cor1} implies immediately a weaker version of Theorem~\ref{prop2}: there exists a Bernstein set $A$ for which each set $p\cap(A\times A)$ has cardinality less than $\continuum$. However, we see no easy way to deduce the full version of the theorem from the results presented above. Nevertheless we like to include here a very short proof of Theorem~\ref{prop2}, since it is considerably simpler and completely different that the argument presented in~\cite{BH}. \noindent \pf First notice that if $A$ is a transcendental base of $\mathR$ (over $\mathQ$) then the set $p\cap(A\times A)$ is finite for every polynomial $p$ which is neither constant nor the identity function. Indeed, if $K\in[A]^{<\omega}$ is such that $p\in\overline{\mathQ(K)}[x]$, where $\overline{\mathQ(K)}$ stands for the algebraic closure of $\mathQ(K)$ in $\real$, then for every $a\in A\setminus K$ we have $p(a)\in\overline{\mathQ(K\cup\{a\})}\setminus\overline{\mathQ(K)}$, since $A$ is algebraically independent. (See e.g., \cite[Lemma~2, p. 99]{MK}.) So, if $p(a)\in A$ then $p(a)=a$. But this is impossible, since $p(a)=a$ implies that $a$ is a root of a non-zero polynomial $p-\id\in\overline{\mathQ(K)}[x]$. So $\pi[p\cap(A\times A)]\subset K$. It is well known that there are transcendental bases $A$ that are also Bernstein sets (in \cite{CiBook} see corollary 7.3.6 and exercise 2 on page 126) and any such base satisfies Proposition~\ref{prop2}. \qed \section{Proof of Theorem~\ref{tmain}} Let $\{\la f_{\aaa},B_{\aaa}\ra\colon\aaa<\co\}$ be an enumeration of $\F\times\B_0$. For each $\aaa<\co$ we will choose, by induction on $\alpha<\co$, points $x_{\aaa}\in B_{\aaa}$ and $c_{\aaa}\in B_{\aaa}$ aiming for $A=\{ x_{\aaa}\colon\aaa<\co\}$. We will set up the induction in such a way that for every $\aaa<\co$ the set $Z$ satisfying (a) for $f_{\aaa}$ will be equal to $A_{\aaa}=\{ x_{\bbb}\colon \bbb<\aaa\}$ and, if (\ref{eqMain2}) holds for every $f\in(\pm\F)\cup(\id-\F)$, that $(f_{\aaa}+c_{\aaa})\cap (A\times A)=\emptyset$. So, assume that for some $\alpha<\continuum$ the sets $\{ x_{\bbb}\colon \bbb<\aaa\}$ and $\{c_{\bbb}\colon \bbb<\aaa\}$ are already constructed. If we need only to insure (a), we put $c_\alpha=x_\alpha$ and choose an $x_{\aaa}\in B_{\aaa}\setminus \bigcup_{\beta\leq\alpha}L_{f_\beta}$ such that \[ f_\beta\cap(\{x_\alpha\}\times (A_\alpha\setminus A_\beta))=\emptyset= f_\beta\cap(A_\alpha\times\{x_\alpha\}) \] for every $\beta\leq\alpha$. This is possible by (\ref{eqMain1}), since $\{L_{f_\beta}\colon\beta\leq\alpha\}\subset\J$, singletons $f_{\bbb}[\{a\}]\in\J$, for $\beta\leq\alpha$ and $a\in A_\alpha$, and $\{x\in\mathR\colon f_{\bbb}(x)=x_\gamma\}=f_{\bbb}^{-1}(x_\gamma)\in\J$ for every $\beta\leq\gamma<\alpha$ since $x_\gamma\notin L_{f_\beta}$. It is easy to see that such a choice implies that $f_\alpha\cap(A\times A)\subset(A\times A_\alpha)\cup\id$ for every $\alpha<\co$. So, in what follows we assume that (\ref{eqMain2}) holds for every $f\in(\pm\F)\cup(\id-\F)$. We assume also that the following inductive conditions hold for every $\alpha<\co$. \begin{description} \item[(I)] $(\id-f_{\aaa})^{-1}(c_{\aaa}) \cup f_\alpha^{-1}(x_\beta-c_\alpha)\in\J$ for every $\beta<\alpha$, and \item[(II)] $f_{\bbb}^{-1}(x_\alpha) \cup f_{\bbb}^{-1}(x_\alpha-c_{\bbb})\in\J$ for every $\beta\leq\alpha$. \end{description} First choose a $c_{\aaa}\in B_{\aaa} \setminus \left(L_{\id-f_\alpha}\cup\bigcup_{\beta<\alpha}(x_\beta+L_{-f_\alpha})\right)$ outside the set \[ \{c\in\real\colon (f_{\aaa}+c)\cap (A_{\aaa}\times A_{\aaa})\neq\emptyset\} =A_{\aaa}-f_{\aaa}[A_{\aaa}]\in [\mathR]^{<\co}. \] Such a choice is possible by the assumption (\ref{eqMain1}), since the singletons and the sets $L_{\id-f_\alpha}$ and $x_\beta+L_{-f_\alpha}$ belong to $\J$. Clearly $c_\alpha\notin L_{\id-f_\alpha}$ insures that $(\id-f_{\aaa})^{-1}(c_{\aaa})\in\J$. Similarly, $c_\alpha\notin x_\beta+L_{-f_\alpha}$ implies that $c_\alpha-x_\beta\notin L_{-f_\alpha}$ so the set $f_\alpha^{-1}(x_\beta-c_\alpha)=(-f_\alpha)^{-1}(c_\alpha-x_\beta)$ belongs to $\J$. So, condition (I) is satisfied. Note also that $c_\alpha\notin A_{\aaa}-f_{\aaa}[A_{\aaa}]$ implies that \begin{description} \item[(i)] $(f_{\aaa}+c_{\aaa})\cap (A_{\aaa}\times A_{\aaa})=\emptyset$. \end{description} Next choose an $x_{\aaa}\in B_{\aaa}\setminus \bigcup_{\beta\leq\alpha}(L_{f_\beta}\cup(c_\beta+L_{f_\beta}))$ such that \begin{equation}\label{eq22} (f_\beta+c_\beta)\cap(\{x_\alpha\}\times A_\alpha)=\emptyset= (f_\beta+c_\beta)\cap(A_\alpha\times\{x_\alpha\}), \end{equation} $x_\alpha\notin\bigcup_{\gamma\leq\alpha}(\id-f_\gamma)^{-1}(c_\gamma)$, and \begin{equation}\label{eq44} f_\beta\cap(\{x_\alpha\}\times (A_\alpha\setminus A_\beta))=\emptyset= f_\beta\cap(A_\alpha\times\{x_\alpha\}) \end{equation} for every $\beta\leq\alpha$. This is possible by (\ref{eqMain1}), since $\{L_{f_\beta}\cup(c_\beta+L_{f_\beta})\colon\beta\leq\alpha\}\subset\J$ and by inductive assumptions (I)\&(II) for every $a\in A_\alpha$ and $\beta\leq\gamma<\alpha$ the following sets belong to $\J$. \begin{itemize} \item $\{x\in\mathR\colon(f_{\bbb}+c_{\bbb})(x)=a\}=f_{\bbb}^{-1}(a-c_{\bbb})$; \item $(f_{\bbb}+c_{\bbb})[\{a\}]$; \item $(\id-f_\gamma)^{-1}(c_\gamma)$; \item $\{x\in\mathR\colon f_{\bbb}(x)=x_\gamma\}=f_{\bbb}^{-1}(x_\gamma)$; \item $f_{\bbb}[\{a\}]$. \end{itemize} Note that $x_{\aaa}\notin\bigcup_{\beta\leq\alpha}L_{f_\beta}$ insures that $f_{\bbb}^{-1}(x_\alpha)\in\J$ and $x_{\aaa}\notin\bigcup_{\beta\leq\alpha}(c_\beta+L_{f_\beta})$ implies that $x_\alpha-c_\beta \notin L_{f_\beta}$ so the set $f_{\bbb}^{-1}(x_\alpha-c_{\bbb})$ is in $\J$. In particular, (II) is satisfied. This finishes the inductive construction. Clearly $A$ is nowhere meager, since it meets every non-meager Borel set. To see that (b) holds notice that by (i) we have $(f_\beta+c_\beta)\cap (A_\beta\times A_\beta)=\emptyset$ while $(f_\beta+c_\beta)\cap (A_{\aaa}\times A_{\aaa})=\emptyset$ for $\alpha>\beta$ is insured by the choice of $x_\alpha$ as in (\ref{eq22}) and the fact that $(f_\beta+c_\beta)(x_\alpha)\neq x_\alpha$ since $x_\alpha\notin(\id-f_\beta)^{-1}(c_\beta)$. To see that (a) holds pick an $f\in\F$ and let $\beta<\co$ be such that $f=f_\beta$. The choice of $x_\alpha$ for $\alpha>\beta$ as in (\ref{eq44}) implies that $f\cap(A\times A)\subset(A\times A_\beta)\cup\id$. \qed \section{Set $A$ from CPA}\label{sec:CPA} To formulate axiom \psmCGame\ we need a few definitions. Let $\Cantor$ denote the Cantor set $2^\omega$. For a Polish space $X$ we use symbol $\perf(X)$ to denote the family of all subsets of $X$ homeomorphic to $\Cantor$. A subset $C$ of a product $\Cantor^\omega$ of the Cantor set is said to be a {\em perfect cube}\/ if $C=\prod_{n\in\omega} C_n$, where $C_n\in\perf(\Cantor)$ for each $n$. For a fixed Polish space $X$ let $\Fcube$ stand for the family of all continuous injections from a perfect cube $C\subset\Cantor^\omega$ onto a set $P$ from $\perf(X)$. We consider each function $f\in\Fcube$ from $C$ onto $P$ as a coordinate system imposed on~$P$. We say that $P\in\perf(X)$ is a {\em cube}\/ if we consider it with (implicitly given) witness function $f\in\Fcube$ onto $P$, and $Q$ is a {\em subcube of a cube}\/ $P\in\perf(X)$ provided $Q=f[C]$, where $f\in\Fcube$ is a witness function for $P$ and $C\subset\dom(f)\subset\Cantor^\omega$ is a perfect cube. (Here and in what follows symbol $\dom(f)$ stands for the domain of $f$.) We say that a family $\E\subset\perf(X)$ is {\em cube dense}\/ in $\perf(X)$ provided every cube $P\in\perf(X)$ contains a subcube $Q\in\E$. More formally, $\E\subset\perf(X)$ is cube dense provided \begin{equation}\label{eqDefFdense} \forall f\in\Fcube\ \exists g\in\Fcube\ (g\subset f\ \&\ \range(g)\in\E). \end{equation} We need also a notion of a {\em constant cubes}: a family $\Ccube(X)$ of constant ``cubes'' is defined as a family of all constant functions from a perfect cube $C\subset\Cantor^\omega$ to $X$. We define $\Fcube^*(X)$ as \begin{equation}\label{eqFcubeSTAR} \Fcube^*=\Fcube\cup\Ccube. \end{equation} Thus, $\Fcube^*$ is the family of all continuous functions from a perfect cube $C\subset\Cantor^\omega$ into $X$ which are either one-to-one or constant. Now the range of every $f\in \Fcube^*$ belongs to the family $\perf^*(X)$\label{PagePerfStar} of all sets $P$ such that either $P\in\perf(X)$ or $P$ is a singleton. The terms ``$P\in\perf^*(X)$ is a cube'' and ``$Q$ is a subcube of a cube $P\in\perf^*(X)$'' are defined in a natural way. \medskip For a Polish space $X$ consider the following game ${\rm GAME}_{\rm cube}(X)$\label{PageGameCube} of length $\omega_1$. The game has two players, Player~I and Player~II. At each stage $\xi<\omega_1$ of the game Player~I can play an arbitrary cube $P_\xi\in\perf^*(X)$ and Player~II must respond with a subcube $Q_\xi$ of $P_\xi$. The game $\la\la P_\xi,Q_\xi\ra\colon \xi<\omega_1\ra$ is won by Player~I provided \[ \bigcup_{\xi<\omega_1}Q_\xi=X; \] otherwise the game is won by Player~II. By a strategy for Player~II we will understand any function $S$ such that $S(\la\la P_\eta,Q_\eta\ra\colon \eta<\xi\ra,P_\xi)$ is a subcube of $P_\xi$, where $\la\la P_\eta,Q_\eta\ra\colon \eta<\xi\ra$ is any partial game. (We abuse here slightly the notation, since function $S$ depends also on the implicitly given coordinate functions $f_\eta\colon \Cantor^\omega\to P_\eta$ making each $P_\eta$ a cube.) A game $\la\la P_\xi,Q_\xi\ra\colon \xi<\omega_1\ra$ is played according to a strategy $S$ provided $Q_\xi=S(\la\la P_\eta,Q_\eta\ra\colon \eta<\xi\ra,P_\xi)$ for every $\xi<\omega_1$. A strategy $S$ for Player~II is a {\em winning strategy}\/ for Player~II provided Player~II wins any game played according to the strategy $S$. Now we can formulate the axiom. (See~\cite{CPAbook}.) \begin{description} \item[{\bf \psmCGame:}] $\continuum=\omega_2$ and for any Polish space $X$ Player~II has no winning strategy in the game ${\rm GAME}_{\rm cube}(X)$. \end{description} All we need to know about cube-dense families is the following fact. \begin{Fact}\label{fact:cube} Let $X$ be a Polish space and let $\E\subset\perf^*(X)$ contain all singletons. If for every $P\in\perf(X)$ and every Borel probability measure $\mu$ on $P$ there exists a $Q\in\perf(P)\cap\E$ such that $\mu(Q)>0$, then $\E$ is cube-dense. \end{Fact} \pf This follows immediately from \cite[Claim~3.2]{CP88}. (See also \cite[Claim~1.1.5]{CPAbook} or \cite[Claim~2.3]{CP83}.) \qed We will apply this fact to $X=\C$, where $\C$ is considered with the sup norm. Notice that for every $Q\subset\C$ the set $\bigcup Q\subset\real^2$ is a union of the graphs of all functions belonging to $Q$, since functions are identified with their graphs. In what follows for a set $K\subset\real^2$ and $x\in\real$ we will use the symbol $K_x$ to denote the vertical section of $K$ above $x$, that is, $K_x=\{y\colon\la x,y\ra\in K\}$. Similarly, $K^x=\{x\colon\la x,y\ra\in K\}$. For $A\in[\real]^{\leq\omega}$ let $\E(A)$ be the family of all $Q\in\perf^*(\C)$ such that \begin{itemize} \item $\bigcup Q$ is nowhere dense in $\real^2$, and \item $[\bigcup Q]_x$ is nowhere dense in $\real$ for every $x\in A$. \end{itemize} \begin{lem}\label{lemCPA} Family $\E(A)$ is cube-dense for every $A\in[\real]^{\leq\omega}$. \end{lem} \pf Without loss of generality we can assume that $A$ is dense in $\real$. Clearly every singleton belongs to $\E(A)$. So, let $P\in\perf(\C)$ and let $\mu$ be a Borel probability measure $\mu$ on $P$. By Fact~\ref{fact:cube} it is enough to show that there exists a $Q\in\perf(P)\cap\E(A)$ such that $\mu(Q)>0$. To see this, fix a countable base $\B$ for $\real$ and let $\la \la a_n,J_n\ra\colon n<\omega\ra$ be an enumeration of $A\times \B$. Notice that for every $n<\omega$ there exists a non-empty open set $U_n\subset J_n$ such that \begin{equation}\label{eq:CPA1} \mu(\{f\in P\colon f(a_n)\in U_n\})<2^{-(n+2)}. \end{equation} Indeed, if $\U_n$ is an infinite family of non-empty pairwise disjoint open subsets of $J_n$ then for each $U\in\U_n$ the set $\{f\in P\colon f(a_n)\in U\}$ is open in $P$ (so $\mu$-measurable) and so condition (\ref{eq:CPA1}) must hold for some $U\in\U_n$. Let $W=\bigcup_{n<\omega} \{f\in\C\colon f(a_n)\in U_n\}$. It is clear that $W$ is open and dense in $\C$. So, $Q=P\setminus W=P\setminus \bigcup_{n<\omega} \{f\in P\colon f(a_n)\in U_n\}$ is nowhere dense (and therefore $\bigcup Q$ is nowhere dense in $\mathR^2$) and, by (\ref{eq:CPA1}), it has $\mu$ measure at least $1-\sum_{n<\omega} 2^{-(n+2)}=2^{-1}>0$. It is also clear that for every $x\in A$ the set $\bigcup\{U_n\colon a_n=x\}$ is dense open in $\real$ and it is disjoint with $[\bigcup Q]_x$. Thus $Q\in\E(A)$. \qed \begin{prop}\label{propCPA} Assume that \psmCGame\ holds, let $X$ be a Polish space, and let $S$ be a mapping associating to every $\bar P\in \bigcup_{\alpha<\omega_1}(\perf^*(X))^\alpha$ a cube-dense family $\E(\bar P)\subset \perf^*(X)$. Then there exists a sequence $\la \la P_\xi,Q_\xi\ra\colon\xi<\omega_1\ra$ such that for every $\xi<\omega_1$ we have $Q_\xi\in\perf^*(P_\xi)\cap\E(\la P_\zeta\colon\zeta<\xi\ra)$ and $X=\bigcup_{\xi<\omega_1}Q_\xi$. \end{prop} \pf This follows easily from \psmCGame. More precisely, it is enough to apply \psmCGame\ to the strategy $S^*$ such that $S^*(\la\la P_\eta,Q_\eta\ra\colon \eta<\xi\ra,P_\xi)$ is a subcube of $P_\xi$ from $S(\la P_\eta\colon \eta<\xi\ra)$. \qed \noindent {\sc Proof of Theorem~\ref{thmCPA}.} First recall that \psmCGame\ implies that the cofinality of the ideal of meager sets is equal to $\omega_1<\continuum$, that is, there exists an $\M_0\in[\M]^{\omega_1}$ such that every meager set is contained in some $M\in\M_0$. (See e.g. \cite[sec.~4]{CP88} or \cite{CPAbook}.) Let $\B_0$ be a countable base for $\real$ and let $\{\la M_\xi,J_\xi\ra\colon\xi<\omega_1\}$ be an enumeration of $\M_0\times \B_0$. By simultaneous induction on $\xi<\omega_1$, using Lemma~\ref{lemCPA}, we will define functions $S$, $Q$, and $k$ on $(\perf^*(\C))^\xi$ such that \begin{itemize} \item[(i)] $S(\la P_\zeta\colon\zeta<\xi\ra)=\E(\{a_\zeta\colon\zeta<\xi\})$, where $a_\zeta=k(\la P_\eta\colon\eta\leq\zeta\ra)\in\real$, and $Q_\xi=Q(\la P_\zeta\colon\zeta<\xi\ra)\in\E(\{a_\zeta\colon\zeta<\xi\})$. \item[(ii)] $k(\la P_\zeta\colon\zeta\leq\xi\ra)$ belongs to $J_\xi$ and the residual set \[ \bigcap_{\zeta\leq\xi}\{z\in\real\colon \mbox{ $(\bigcup Q_\zeta)^z$ and $(\bigcup Q_\zeta)_z$ are nowhere dense in $\real$}\}. \] \item[(iii)] $k(\la P_\zeta\colon\zeta\leq\xi\ra)$ does not belong to the meager set \[ M_\xi\cup \bigcup_{\eta\leq\xi} \mbox{$(\{(\bigcup Q_\eta)_{a_\zeta}\colon\zeta<\xi\}\cup \{(\bigcup Q_\eta)^{a_\zeta}\colon\eta\leq\zeta<\xi\})$}. \] \end{itemize} The set as in (ii) is residual by Kuratowski-Ulam theorem, since each set $\bigcup Q_\zeta$ is nowhere dense, as $Q_\zeta$ belongs to some $\E(A)$. In (iii) for every $\eta\leq\zeta<\xi$ the set $(\bigcup Q_\eta)_{a_\zeta}\cup (\bigcup Q_\eta)^{a_\zeta}$ is nowhere dense by the choice of $a_\zeta=k(\la P_\eta\colon\eta\leq\zeta\ra)$ as in (ii). Finally, for $\zeta<\eta$ the set $(\bigcup Q_\eta)_{a_\zeta}$ is nowhere dense since, by (i), $Q_\eta$ belongs to $S(\la P_\zeta\colon\zeta<\eta\ra)=\E(\{a_\zeta\colon\zeta<\eta\})$. Now, by the axiom \psmCGame\ and Proposition~\ref{propCPA}, there exists a sequence $\la \la P_\xi,Q_\xi,a_\xi\ra\colon\xi<\omega_1\ra$ such that $\C=\bigcup_{\xi<\omega_1}Q_\xi$ and the conditions (i)--(iii) are satisfied. We claim that $A=\{a_\xi\colon\xi<\omega_1\}$ satisfies Theorem~\ref{thmCPA}. Clearly, $A$ is nowhere meager since for every non-empty open set $U\subset\real$ and every meager set $M$ there exists a $\xi<\omega_1$ such that $J_\xi\subset U$ and $M\subset M_\xi$. But then $a_\xi\in (A\cap J_\xi)\setminus M_\xi\subset(A\cap U)\setminus M$, so $A\cap U\neq M$. To see the main part of Theorem~\ref{thmCPA} take an $f\in\C$. Then, there exists a $\eta<\omega_1$ such that $f\in Q_\eta$. We claim that for $Z=\{a_\beta\colon \beta<\eta\}$ we have $f\cap(A\times A)\subset(A\times Z)\cup\id$. Indeed, let $\eta\leq\xi<\omega_1$ and $\zeta<\omega_1$ be such that $\zeta\neq\xi$. We need to show that $\la a_\zeta,a_\xi\ra\notin f$. But if $\zeta<\xi$ then, by (iii), $a_\xi$ does not belong to $[\bigcup Q_\eta]_{a_\zeta}\ni f(a_\zeta)$, so $\la a_\zeta,a_\xi\ra\notin f$. Similarly, if $\xi<\zeta$ then, again by (iii), $a_\zeta$ does not belong to $[\bigcup Q_\eta]^{a_\xi}\supset f^{-1}(a_\xi)$ and once more $\la a_\zeta,a_\xi\ra\notin f$. \qed \section{Main consistency result}\label{sec:oracle} The main goal of this section is to prove the following theorem. \begin{thm}\label{thConsOracle} It is relatively consistent with ZFC that $\continuum=\omega_2$ and the following two conditions hold simultaneously. \begin{description} \item[(A)] For every family $\{B_\xi\colon\xi<\omega_1\}$ of pairwise disjoint nowhere meager subsets of $\real^2$ there exists an increasing homeomorphism $f\colon\real\to\real$ such that $\pi[f\cap B_\xi]$ is nowhere meager for every $\xi<\omega_1$. \item[(B)] Every nowhere meager set $B\subset \real$ contains a nowhere meager subset of cardinality $\omega_1$. \end{description} \end{thm} We use Theorem~\ref{thConsOracle}, in conjunction with the following proposition, to deduce Theorem~\ref{thmCons1}. \begin{prop}\label{propA} Assume that $\continuum>\omega_1$ and that {\rm (A)} and\/ {\rm (B)} hold. Then for every nowhere meager set $A\subset\mathR$ there is a homeomorphism $f\colon\real\to\real$ such that $\{c\in\real\colon \pi[(f+c)\cap(A\times A)]\notin\M\}$ is nowhere meager. \end{prop} \pf By condition (B) we can assume that $|A|=\omega_1<\continuum$. Let $B$ be a Bernstein set such that $(b+A)\cap(b'+A)=\emptyset$ for every distinct $b,b'\in B$. Let $\{c_\xi\colon\xi<\omega_1\}$ be a nowhere meager subset of $B$ and let $B_\xi=A\times(c_\xi+A)$ for $\xi<\omega_1$ and let $f\colon\real\to\real$ be as in (A). Then set $\{c\in\real\colon \pi[(f+c)\cap(A\times A)]\notin\M\}$ contains a nowhere meager set $\{-c_\xi\colon\xi<\omega_1\}$ since for every $\xi<\omega_1$ we have \[ \pi[(f-c_\xi)\cap(A\times A)]=\pi[f\cap B_\xi]\notin\M \] finishing the proof. \qed The proof of Theorem~\ref{thConsOracle} is a slight modification of the proof of the main result (Theorem 2) from~\cite{CS2}. Also, Theorem~\ref{thConsOracle} easily implies \cite[Theorem 2]{CS2}. We will use here terminology and notation from \cite{CS2}. In particular, according to the machinery used in this paper, Theorem~\ref{thConsOracle} follows in a standard way from the following lemma. (More precisely, condition (A) is ensured by the lemma, while (B) and $\continuum=\omega_2$ are guaranteed by the iteration procedure.) \begin{lem}\label{lemMain} For every family $\B=\{B_\xi\colon\xi<\omega_1\}$ of pairwise disjoint nowhere meager subsets of $\real^2$ and for every $\omega_1$-oracle $\M$ there exists an $\M$-cc forcing notion $Q_\B$ of cardinality $\omega_1$ such that $Q_\B$ forces \begin{quote} there exists an increasing homeomorphism $f\colon\real\to\real$ such that $\pi[f\cap B_\xi]$ is nowhere meager for every $\xi<\omega_1$. \end{quote} \end{lem} In what follows we will present the proof of Lemma~\ref{lemMain}. Let \[ \Gamma=\{\lambda<\omega_1\colon \lambda \mbox{ is a limit ordinal}\}. \] Recall that an {\em $\omega_1$-oracle}\/ is any sequence $\M=\la M_{\delta}\colon \delta\in\Gamma\ra$, where $M_{\delta}$ is a countable transitive model of ZFC$^{-}$ (that is, ZFC without the power set axiom) with a property that $\delta+1\subset M_{\delta}$, $\delta$ is countable in $M_{\delta}$, and the set $\{\delta\in\Gamma\colon A\cap\delta\in M_{\delta}\}$ is stationary in $\omega_1$ for every $A\subset\omega_1$. With each $\omega_1$-oracle $\M=\la M_\delta\colon\delta\in\Gamma\ra$ there is associated a filter $D_{\M}$ generated by the sets $I_\M(A)=\{\delta\in\Gamma\colon A\cap\delta\in M_\delta\}$ for $A\subset\omega_1$. It is proved in~\cite[Claim~1.4]{Sh:f} that $D_{\M}$ is a proper normal filter containing every closed unbounded subset of $\Gamma$. We will also need the following fact which, for our purposes, can be viewed as a definition of $\M$-cc property. \fact{f1}{{\rm (\cite[Fact~4]{CS2})} Let $P$ be a forcing notion of cardinality $\leq\omega_1$, $e\colon P\to\omega_1$ be one-to-one, and $\M=\la M_\delta\colon\delta\in\Gamma\ra$ be an $\omega_1$-oracle. If there exists a $C\in D_\M$ such that for every $\delta\in\Gamma\cap C$ \begin{quote} $e^{-1}(E)$ is predense in $P$ for every set $E\in M_\delta\cap\P(\delta)$, for which $e^{-1}(E)$ is predense in $e^{-1}(\{\gamma\colon \gamma<\delta\})$, \end{quote} then $P$ has the $\M$-cc property. \qed} Let $\K$ be the family of all sequences $\bar h=\la h_\xi\colon\xi\in\Gamma\ra$ such that each $h_\xi$ is a function from a countable set $D_\xi\subset\real$ onto $R_\xi\subset \real$ such that $h_\xi$ is dense in $\real^2$ and \[ D_\xi\cap D_\eta=R_\xi\cap R_\eta=\emptyset \ \mbox{ for every distinct }\ \xi,\eta\in\Gamma. \] For each $\bar h\in\K$ we will define a forcing notion $Q_{\bar h}$. Forcing $Q_\B$ satisfying Lemma~\ref{lemMain} will be chosen as $Q_{\bar h}$ for some $\bar h\in\K$. So let $\H$ be the family of all strictly increasing functions from finite subsets of $\real$ into $\real$ and fix an $\bar h\in\K$. Then $Q_{\bar h}$ is defined as \[ Q_{\bar h}=\left\{h\in\H\colon h\subset\bigcup_{\xi\in\Gamma}h_\xi \ \&\ |h\cap h_\xi|\leq 1\mbox{ for every }\xi\in\Gamma\right\} \] and is ordered by the reverse inclusion. In what follows we will use the following basic property of $Q_{\bar h}$. \fact{fact2}{Let $\bar h=\la h_\xi\colon\xi\in\Gamma\ra\in\K$ and $f=\bigcup H$, where $H$ is a $V$-generic filter over $Q_{\bar h}$. Then $f$ is a strictly increasing function from a dense subset $D$ of $\real$ onto a dense subset of $\real$. In particular, $f$ can be uniquely extended to an increasing homeomorphism $\tilde f$ of $\real$. } \pf Clearly $f$ is a strictly increasing function from a subset $D$ of $\real$ onto a subset $R$ of $\real$. Thus, it is enough to show that $D$ and $R$ are dense in $\real$. So, let $U\neq\emptyset$ be open in $\real$ and notice that the set \[ D=\{h\in Q_{\bar h}\colon\dom(h)\cap U\neq\emptyset\} \] is dense in $Q_{\bar h}$. Indeed, if $h_0\in Q_{\bar h}$ is such that $\dom(h_0)\cap U=\emptyset$ then we can find $\xi\in\Gamma$ such that $h\cap h_\xi=\emptyset$. Since graph of $h_\xi$ is dense in $\real^2$ we can find $\la x,y\ra\in h_\xi$ such that $x\in U$ and $h=h_0\cup\{\la x,y\ra\}$ is strictly increasing. Then $h\in D$ extends $h_0$. Similarly we can proved that the set \[ \{h\in Q_{\bar h}\colon\range(h)\cap U\neq\emptyset\} \] is dense in $Q_{\bar h}$. The rest follows from the genericity of $H$. \qed Now let $\B=\{B_\xi\colon\xi<\omega_1\}$ be as in the lemma and fix an $\omega_1$-oracle $\M=\la M_\delta\colon\delta\in\Gamma\ra$. By Fact~\ref{fact2} in order to prove Lemma~\ref{lemMain} it is enough to find an $\bar h=\la h_\xi\colon\xi\in\Gamma\ra\in\K$ such that \begin{equation}\label{eqN2} \mbox{$Q_\B=Q_{\bar h}$ is $\M$-cc} \end{equation} and $Q_{\bar h}$ forces that, in $V[H]$, \begin{equation}\label{eqN3} \mbox{the sets $\pi(f\cap B_\xi)$ is nowhere meager for every $\xi<\omega_1$,} \end{equation} where function $f$ is defined as in Fact~\ref{fact2}. To define $\bar h$ we will construct a sequence $\la \la x_\alpha,y_\alpha\ra\in \real^2\colon \alpha<\omega_1\ra$ aiming at $h_\xi=\{\la x_{\xi+n},y_{\xi+n}\ra\colon n<\omega\}$, where $\xi\in\Gamma$. So, let $\U\not\ni\emptyset$ be a standard countable basis for $\real$ and for every $\xi\in\Gamma$ let $\la\la U^\xi_n,V^\xi_n,\zeta^\xi_n\ra\colon n<\omega\ra$ be a fixed enumeration of $\U\times\U\times\xi$. Points $\la x_{\xi+n},y_{\xi+n}\ra$ are chosen inductively in such a way that \begin{description} \item{(i)} $\la x_{\xi+n},y_{\xi+n}\ra$ is a Cohen real over $M_\delta[\la \la x_\alpha,y_\alpha\ra\colon \alpha<\xi+n\ra]$ for every $\delta\leq\xi$, $\delta\in\Gamma$, that is, $\la x_{\xi+n},y_{\xi+n}\ra$ is outside all meager subsets of $\real^2$ which are coded in $M_\delta[\la \la x_\alpha,y_\alpha\ra\colon \alpha<\xi+n\ra]$; \item{(ii)} $\la x_{\xi+n},y_{\xi+n}\ra\in(U^\xi_n\times V^\xi_n)\cap B_{\zeta^\xi_n}$. \end{description} The choice of $\la x_{\xi+n},y_{\xi+n}\ra$ is possible since the sets $(U^\xi_n\times V^\xi_n)\cap B_{\zeta^\xi_n}$ are non-meager and each time we need to avoid only countably many meager sets. Condition (ii) guarantees that the graph of each of $h_\xi$ will be dense in $\real^2$. Note also that if $\Gamma\ni\delta\leq \alpha_0<\cdots<\alpha_{k-1}$, where $k<\omega$, then (by the product lemma in $M_\delta$) \begin{equation}\label{PLem} \mbox{$\la\la x_{\alpha_i},y_{\alpha_i}\ra\colon i\zeta$ from $B_0$ and let $U,W\in\U$ be such that $U\subset U^*$ and $h^*\cup\{\la x,y\ra\}\in\H$ for every $\la x,y\ra\in U\times W$. Using clause (ii) of the choice of $x_\alpha$'s we may find an $n<\omega$ such that $\la x_{\delta+n},y_{\delta+n}\ra\in(U\times W)\cap B_\zeta$. Then $h_0=h^*\cup\{\la x_{\delta+n},y_{\delta+n}\ra\}\in Q_{\bar{h}}$ extends $h^*$ and $h_0\forces``x_{\delta+n}\in U^*\cap\pi(f\cap B_\zeta)$'' since $\langle x_{\delta+n},y_{\delta+n}\rangle\in f$. We will show that \[ h_0\forces x_{\delta+n}\in \bigcap_{m<\omega}\dot{U}_m, \] which contradicts the choice of $h^*$. So, assume that this is not the case. Then there exist an $i<\omega$ and an $h_1\in Q_{\bar h}$ stronger than $h_0$ such that $h_1\forces``x_{\delta+n}\notin\dot{U}_i$.'' Let us define $h=h_1\restriction\{x_\alpha\colon \alpha<\delta\}\in (Q_{\bar h})^\delta$ and $h_1\setminus h=\{\la a_l,b_l\ra\colon l