Observation 1.1. If $\mathcal {I}\ne \mathrm {Fin}$ , $\mathcal {J}$ is a P-ideal, and $\mathcal {I}\leq _{\mathrm {K}}\mathcal {J}$ , then $\mathcal {I}\leq _{\mathrm {KB}}\mathcal {J}$ holds as well.

Proof. Fix a K-reduction $f:\omega \to \omega $ (that is, $f^{-1}[A]\in \mathcal {J}$ for every $A\in \mathcal {I}$ ) which is not finite-to-one and a $B\in \mathcal {J}$ such that $f^{-1}[\{n\}]\subseteq ^* B$ for every n (in particular, B is infinite). Let $F_n=f^{-1}[\{n\}]\setminus B$ and fix an infinite element $A\in \mathcal {I}$ . Define $g:\omega \to \omega $ such that $g\!\upharpoonright \! F_n\equiv n$ for every n and $g\!\upharpoonright \! B$ is a bijection between B and A. Then g is a KB-reduction.⊣

Fact 2.3. Assume that $\mathcal {I}$ is a nowhere tall analytic P-ideal. Then $\mathcal {I}$ is a trivial modification of $\mathrm {Fin}$ or $\mathcal {I}\simeq \{\varnothing \}\otimes \mathrm {Fin}$ .

Proof. Let $\mathcal {I}=\mathrm {Exh}(\varphi )$ for some lsc submeasure $\varphi $ . First we show that $\mathcal {I}$ is nowhere tall iff $\mathcal {I}=\mathcal {I}':=\{A\subseteq \omega :A$ is finite or $\lim _{n\in A}\varphi (\{n\})=0\}$ .

First assume that $\mathcal {I}$ is nowhere tall. As $\mathcal {I}\subseteq \mathcal {I}'$ always holds, we show that if $A\in \mathcal {I}'$ then $A\in \mathcal {I}$ . Assume on the contrary that $A\notin \mathcal {I}$ . Then there is an infinite $A'\subseteq A$ such that $\mathcal {I}\!\upharpoonright \! A'=[A']^{<\omega }$ . If $B=\{b_0<b_1<b_2<\cdots \}\subseteq A'$ such that $\varphi (\{b_n\})<2^{-n}$ for every n, then $B\in \mathcal {I}$ , a contradiction. Conversely, assume now that $\mathcal {I}=\mathcal {I}'$ , and let $X\in \mathcal {I}^+$ . If $Y\subseteq X$ is infinite and $\inf \{\varphi (\{k\}):k\in Y\}>0$ then $\mathcal {I}\!\upharpoonright \! Y=[Y]^{<\omega }$ and hence $\mathcal {I}\!\upharpoonright \! X$ is not tall.

Therefore, if $\mathcal {I}$ is a nowhere tall analytic P-ideal, then there is a sequence $x_n>0$ such that $\mathcal {I}=\{A\subseteq \omega :A$ is finite or $\lim _{n\in A}x_n=0\}$ . By slightly perturbing $x_n$ , we can assume that $x_n\ne x_m$ for every $n\ne m$ . Let $X=\{x_n:n\in \omega \}$ , $X'$ be set of accumulation points of X, and $X"=(X')'$ ; we know that $X'\supseteq X"$ are closed. We have the following three cases: (1) $0\notin X'$ . Then $\mathcal {I}=\mathrm {Fin}$ . (2) $0\in X'\setminus X"$ . Let $y=\min (X'\setminus \{0\})>0$ . Now $A\subseteq \omega $ belongs to $\mathcal {I}$ iff $|A\cap \{n:x_n\geq y\}|<\omega $ , hence $\mathcal {I}$ is a trivial modification of $\mathrm {Fin}$ . (3) $0\in X"$ . Fix a sequence $y_0=\infty>y_1>y_2\dots $ tending to $0$ such that each $[y_{k+1},y_k)$ contains infinitely many $x_n$ , and let $P_k=\{n:x_n\in [y_{k+1},y_k)\}$ . Then $A\in \mathcal {I}$ iff $A\cap P_k$ is finite for every k, and hence $\mathcal {I}\simeq \{\varnothing \}\otimes \mathrm {Fin}$ .⊣

Proof. (1a): Let us recall the following characterisations of $\mathrm {cov}(\mathcal {M})$ (see [Reference Bartoszyński and Judah3, Lemma 2.4.2 and Theorem 2.4.5]): Let $\mathcal {C}=\{S\in \,\!^\omega ([\omega ]^{<\omega }):\sum _{n\in \omega }|S(n)|/n^2<\infty \}$ (for now let $1/0=1$ ). Then

$$ \begin{align*} \mathrm{cov}(\mathcal{M}) =\min\big\{|F|:F\subseteq\,\!^\omega\omega\;\forall\;S\in\mathcal{C}\;\exists\;f\in F\;\forall^\infty\;n\;f(n)\notin S(n)\big\} \end{align*} $$

$$ \begin{align*}=\min\big\{|\mathcal{D}|:\forall\;D\in\mathcal{D}\;(D\subseteq\mathbb{C}\;\text{is dense})\;\text{and}\;\nexists\;\mathcal{D}\text{-generic filter}\;G\subseteq\mathbb{C}\big\}.\end{align*} $$

To show $\mathrm {non}^*(\mathcal {ED},+)\leq \mathrm {cov}(\mathcal {M})$ , fix an $F\subseteq \,\!^\omega \omega $ witnessing the above characterisation. For every $f\in F$ define $X_f\in \mathcal {ED}^+$ as follows:

$$ \begin{align*}X_f=\big\{(n,k)\in \omega\times\omega:f(n)-\lfloor\sqrt{n}\rfloor\leq k\leq f(n)+\lfloor\sqrt{n}\rfloor\big\}.\end{align*} $$

Let $A\in \mathcal {ED}$ , we show that there is an $f\in F$ such that $|A\cap X_f|<\omega $ , and hence $\mathrm {non}^*(\mathcal {ED},+)\leq |F|=\mathrm {cov}(\mathcal {M})$ . As columns have finite intersection with every $X_f$ , we can assume that A is of the form $\bigcup \{\{n\}\times F_n:n\in \omega \}$ where $F_n\in [\omega ]^m$ for some fixed $m\in \omega $ . Let $S_A:\omega \to [\omega ]^{<\omega }$ ,

$$ \begin{align*} S_A(n)=\bigcup_{k\in F_n}\big[k-\lfloor\sqrt{n}\rfloor, k+\lfloor\sqrt{n}\rfloor\big]\cap\omega.\end{align*} $$

Then $S_A\in \mathcal {C}$ and hence there is an $f\in F$ such that $f(n)\notin S_A(n)$ for almost all n, and so $(\{n\}\times F_n)\cap X_f=\varnothing $ for almost all n, i.e., $|A\cap X_f|<\omega $ .

Conversely, we show that if a family $\{X_\alpha :\alpha <\kappa \}$ witnesses $\mathrm {non}^*(\mathcal {ED},+)$ then there is a family $\mathcal {D}$ of dense subsets of $\mathbb {C}$ , $|\mathcal {D}|=\kappa $ such that no filter on $\mathbb {C}$ is $\mathcal {D}$ -generic. We know that for every $\alpha $ there are infinitely k such that $(X_\alpha )_k\ne \varnothing $ . Interpret $\mathbb {C}$ now as $(\,\!^{<\omega }\omega ,\supseteq )$ and for every $\alpha $ and n let $D_{\alpha ,n}=\{s\in \mathbb {C}:\exists\ k\geq n\ s(k)\in (X_\alpha )_k\}$ . Then $D_{\alpha ,n}$ is dense in $\mathbb {C}$ , and if $G\subseteq \mathbb {C}$ is a $\{D_{\alpha ,n}:\alpha <\kappa ,n\in \omega \}$ -generic filter, then $g=\bigcup G:\omega \to \omega $ (in particular, $g\in \mathcal {ED}$ ) and $|g\cap X_\alpha |=\omega $ for every $\alpha $ , a contradiction.

(1b): We already know that $\mathrm {cov}^*(\mathcal {ED},+)\leq \mathrm {cov}^*(\mathcal {ED},\infty )=\mathrm {non}(\mathcal {M})$ . To show the reverse inequality, we will need the following characterisation (see [Reference Bartoszyński and Judah3, Lemma 2.4.8]):

$$ \begin{align*} \mathrm{non}(\mathcal{M})=\min\big\{|\mathcal{S}|:\mathcal{S}\subseteq\mathcal{C}\;\text{and}\;\forall\;f\in\,\!^\omega\omega\;\exists\;S\in\mathcal{S}\;\exists^\infty\;n\;f(n)\in S(n)\big\}.\end{align*} $$

Notice that there is a family witnessing $\mathrm {cov}^*(\mathcal {ED},+)$ of the form $\{\{n\}\times \omega :n\in \omega \}\cup \{f_\alpha :\alpha <\kappa \}$ where $f_\alpha :\omega \to \omega $ . For every $\alpha $ define $S_\alpha \in \mathcal {C}$ , $S_\alpha (n)=(X_{f_\alpha })_n=\big [f_\alpha (n)-\lfloor \sqrt {n}\rfloor , f_\alpha (n)+\lfloor \sqrt {n}\rfloor \big ]\cap \omega $ . Using the same argument we used in (1a), one can easily show that $\{S_\alpha :\alpha <\kappa \}$ satisfies the conditions in the above characterisation of $\mathrm {non}(\mathcal {M})$ , and hence $\mathrm {non}(\mathcal {M})\leq \kappa $ .

(2): The first “left to right” implication is trivial. The second one is basically [Reference Bartoszyński and Judah3, Lemma 2.4.8, (2) $\to$ (3)]. We show the third. Assume that in an extension $W\supseteq V$ there is an $A\in [\omega \times \omega ]^\omega $ such that $|A\cap B|<\omega $ for every $B\in \mathcal {ED}\cap V$ . By shrinking A can assume that $A=\{(n,k_n):n\in E\}$ , $E\in [\omega ]^\omega $ is an infinite partial function. Let $E=\{n_0<n_1<\cdots \}$ , $\mathrm {FP}=\{$ finite partial functions $\omega \to \omega \}$ , and let $f\in \,\!^\omega \mathrm {FP}\cap W$ , $f(m)=\{(n_i,k_{n_i}):i\leq m\}$ . We show that f is an eventually different real over $\,\!^\omega \mathrm {FP}\cap V$ . Let $g\in \,\!^\omega \mathrm {FP}\cap V$ and assume on the contrary that $f(m)=g(m)$ for infinitely many m. We can assume that $|\mathrm {dom}(g(m))|=m+1$ for every m. Define the infinite partial function $g'\in V$ by recursion as follows: Let $\mathrm {dom}(g(0))=\{m_0\}$ and $g'(m_0)=g(0)(m_0)$ . If we already have $m_0,m_1,\dots ,m_{n-1}$ and $g'$ is defined on these entries, then pick an $m_n\in \mathrm {dom}(g(n))\setminus \{m_0,m_1,\dots ,m_{n-1}\}$ and define $g'(m_n)=g(n)(m_n)$ . It is trivial to show that $|A\cap g'|=\omega $ , a contradiction.

Finally we show that if $f\in \,\!^\omega \omega \cap W$ is an eventually different real over V then $\mathcal {ED}\cap V$ is $+$ -destroyed in W. Fix an interval partition $(P_n)_{n\in \omega }$ in V such that $|P_n|=n+1$ , and fix enumerations $\{(a^n_i,b^n_i):i\in \omega \}=P_n\times \omega $ . Define $X=\{(n,i):f(a^n_i)=b^n_i\}\in \mathcal {ED}^+\cap W$ (because $|(X)_n|=n+1$ ). We claim that $X\ +$ -destroys $\mathcal {ED}\cap V$ . Let $g\in \,\!^\omega \omega \cap V$ and assume on the contrary that $|X\cap g|=\omega $ . Define $g'\in \,\!^\omega \omega \cap V$ as follows: If $g(n)=i$ then let $g' \!\upharpoonright \! P_n\equiv b^n_i$ . It follows that $f(a)=g'(a)$ for infinitely many a, a contradiction.⊣

Proof. (1): For every $n\in \omega $ fix a partition $\{P^n_k:k\leq n\}$ of $(\Delta )_{(n+1)^2-1}=\{((n+1)^2-1,i):i< (n+1)^2\}$ such that $|P^n_k|=n+1$ for every k, and define the following functions:

1. (i) $f:\Delta \to [\Delta ]^{<\omega }$ , $f(n,k)=P^n_k$ ;

2. (ii) $\alpha :[\Delta ]^\omega \to \mathcal {ED}_{\mathrm {fin}}^+$ , $\alpha (X)=\bigcup \{f(n,k):(n,k)\in X\}$ ; and

3. (iii) $\beta :\mathcal {ED}_{\mathrm {fin}}\to \mathcal {ED}_{\mathrm {fin}}$ , $\beta (A)=\{(n,k):f(n,k)\cap A\ne \varnothing \}$ .

We know that $\mathrm {non}^*(\mathcal {ED}_{\mathrm {fin}},+)\geq \mathrm {non}^*(\mathcal {ED}_{\mathrm {fin}},\infty )$ and $\mathrm {cov}^*(\mathcal {ED}_{\mathrm {fin}},+)\leq \mathrm {cov}^*(\mathcal {ED}_{\mathrm {fin}},\infty )$ . Now, if $\mathcal {X}\subseteq [\Delta ]^\omega $ witnesses $\mathrm {non}^*(\mathcal {ED}_{\mathrm {fin}},\infty )$ then $\alpha [\mathcal {X}]=\{\alpha (X):X\in \mathcal {X}\}$ witnesses $\mathrm {non}^*(\mathcal {ED}_{\mathrm {fin}},+)$ : Otherwise, if $A\in \mathcal {ED}_{\mathrm {fin}}$ and $|A\cap \alpha (X)|=\omega $ for every $X\in \mathcal {X}$ , then $|\beta (A)\cap X|=\omega $ for every $X\in \mathcal {X}$ , a contradiction. Similarly, if $\mathcal {A}\subseteq \mathcal {ED}_{\mathrm {fin}}$ witnesses $\mathrm {cov}^*(\mathcal {ED}_{\mathrm {fin}},+)$ then $\beta [\mathcal {A}]$ witnesses $\mathrm {cov}^*(\mathcal {ED}_{\mathrm {fin}},\infty )$ : Otherwise, if $Y\in [\Delta ]^\omega $ has finite intersection with all $\beta (A)$ , then $\alpha (Y)\in \mathcal {ED}_{\mathrm {fin}}^+$ has finite intersection with all $Y\in \mathcal {A}$ , a contradiction.

(2): All “left to right” implications are trivial. Assume now that $V\subseteq W$ is an extension and $g\in W$ is an eventually different infinite partial function over V, $g\leq h\in \omega ^\omega \cap V$ . We can assume that h is strictly increasing. It is straightforward to show that $X=\{(h(n),g(n)):n\in \omega \}\subseteq \Delta $ is also an eventually different infinite partial function over V, and hence $\alpha (X)\ +$ -destroys $\mathcal {ED}_{\mathrm {fin}}\cap V$ .⊣

Proof. (1a): Let $\mathcal {F}=\{F\in [\,\!^{<\omega }2]^{<\omega }\setminus \{\varnothing \}:\sum _{t\in F} 2^{-|t|}\leq 1/4\}$ and for every $F\in \mathcal {F}$ define the clopen set $U_F=\bigcup _{t\in F}\{x\in \,\!^\omega 2:t\subseteq x\}$ and the family $\mathcal {U}_F=\{C\in \Omega :C\cap U_F=\varnothing \}$ . Notice that $\mathcal {U}_F\in \mathcal {S}^+$ because if $X\subseteq \,\!^\omega 2$ is finite then $U_F\cup X$ is a closed set of measure $\leq 1/4<1/2$ and hence there is a clopen set C of measure $1/2$ inside its complement; therefore, $\mathcal {U}_F\nsubseteq \bigcup _{x\in X}\mathcal {C}_x$ . We show that for every $\mathcal {A}\in \mathcal {S}$ there is an $F\in \mathcal {F}$ such that $\mathcal {A}\cap \mathcal {U}_F=\varnothing $ , i.e., that $\{\mathcal {U}_F:F\in \mathcal {F}\}$ witnesses $\mathrm {non}^*(\mathcal {S},+)=\omega $ . Let $\{x_i:i<k\}\subseteq \,\!^\omega 2$ be finite. Pick finite initials $t_i\subseteq x_i$ such that $\sum _{i<k}2^{-|t_i|}\leq 1/4$ and let $F=\{t_i:i<k\}\in \mathcal {F}$ , then $\mathcal {U}_F\cap \bigcup _{i<k}\mathcal {C}_{x_i}=\varnothing $ (because $x_i\in U_F$ for every i, and $C\cap U_F=\varnothing $ for every $C\in \mathcal {U}_F$ ).

(1b): Let $\lambda ^*$ be the Lebesgue outer measure on $\,\!^\omega 2$ . We show that if $\lambda ^*(Y)<1/2$ then there is an $\mathcal {S}$ -positive $\mathcal {D}\subseteq \Omega $ such that $|\mathcal {C}_y\cap \mathcal {D}|<\omega $ for every $y\in Y$ . This implies that $\mathrm {non}(\mathcal {N})\leq \mathrm {cov}^*(\mathcal {S},+)$ , and the reverse inequality follows from $\mathrm {cov}^*(\mathcal {S},+)\leq \mathrm {cov}^*(\mathcal {S},\infty )=\mathrm {non}(\mathcal {N})$ .

Fix an increasing sequence of clopen sets $U_n$ such that $Y\subseteq \bigcup _{n\in \omega }U_n$ and the measure of this union is less than $1/2-\varepsilon $ for some $\varepsilon>0$ . Enumerate $\{V_n:n\in \omega \}$ all clopen sets of measure $<\varepsilon $ and for each n pick a $C_n\in \Omega $ such that $C_n\cap (U_n\cup V_n)=\varnothing $ (this is possible because $U_n\cup V_n$ is a closed set of measure $<1/2$ ). The set $\mathcal {D}=\{C_n:n\in \omega \}$ is $\mathcal {S}$ -positive because if $X\subseteq \,\!^\omega 2$ is finite, then $X\subseteq V_n$ for some n, hence $C_n\notin \bigcup _{x\in X} \mathcal {C}_x$ (and so $\mathcal {D}\nsubseteq \bigcup _{x\in X} \mathcal {C}_x$ ). Also, if $y\in Y$ , then $y\in U_n$ in particular $y\notin C_n$ for every large enough n, and hence $|\mathcal {D}\cap \mathcal {C}_y|<\omega $ .

(2): The first “only if” implication is trivial.

Now assume that $\mathbb {P}$ destroys $\mathcal {S}$ . We will need the following result (see [Reference Meza-Alcántara37, Lemma 1.6.3(b)]): If $\lambda ^*(Y)>1/2$ then for every infinite $\mathcal {D}\subseteq \Omega $ there is a $y\in Y$ such that $|\mathcal {D}\cap \mathcal {C}_y|=\omega $ . This implies that $\Vdash _{\mathbb {P}}\lambda ^*(\,\!^\omega 2\cap V)\leq 1/2$ . Notice that $\Vdash _{\mathbb {P}}$ “ $\lambda ^*(\,\!^\omega 2\cap V)=0$ or $1$ ” holds for every $\mathbb {P}$ : If $V[G]\models \lambda ^*(\,\!^\omega 2\cap V)<1$ then there is a compact set $C\in V[G]$ of positive measure which is disjoint from V, and hence, applying the $0$ – $1$ law, the $F_\sigma $ tail-set $\{x\in \,\!^\omega 2:\exists\ y\in C |x\vartriangle y|<\omega \}$ generated by C is of measure $1$ , and of course, this set is also disjoint from V. We conclude that $\Vdash _{\mathbb {P}} \,\!^\omega 2\cap V\in \mathcal {N}$ .

Finally, the result we proved and applied in (1b) implies that if $\Vdash _{\mathbb {P}} \,\!^\omega 2\cap V\in \mathcal {N}$ then $\mathbb {P}\ +$ -destroys $\mathcal {S}$ .⊣

Proof. (1): A countable base of the topology witnesses $\mathrm {non}^*(\mathrm {Nwd},+)=\omega $ , and reformulating a result from [Reference Keremedis30] (see also in [Reference Balcar, Hernández-Hernández and Hrušák1]) shows that $\mathrm {cov}^*(\mathrm {Nwd},+)=\mathrm {add}(\mathcal {M})$ .

(2): We already know that $\mathbb {C}=\mathbb {P}_{\mathcal {M}}$ destroys $\mathrm {tr}(\mathcal {M})\simeq \mathrm {Nwd}$ , and hence adding a Cohen real destroys $\mathrm {Nwd}$ .

First we show that if $\mathbb {P}\ +$ -destroys $\mathrm {Nwd}$ , then $\mathbb {P}$ adds a dominating real. For now let $\mathrm {Nwd}=\{A\subseteq \,\!^{<\omega } 2:\forall \ s\ \exists \ t\ (s\subseteq t$ and $A\cap t^\uparrow =\varnothing )\}$ , and let $\mathring {X}$ be a $\mathbb {P}$ -name such that $\Vdash _{\mathbb {P}}\mathring {X}\in \mathrm {Nwd}^+$ and $\Vdash _{\mathbb {P}}|\mathring {X}\cap A|<\omega $ for every $A\in \mathrm {Nwd}\cap V$ . We can assume that $\mathring {X}$ is dense in $\,\!^{<\omega } 2$ , that is, $\Vdash _{\mathbb {P}}\forall \ s\ \exists \ t\in \mathring {X} s\subseteq t$ (because there is a $\mathbb {P}$ -name $\mathring {t}$ for a node in $\,\!^{<\omega } 2$ such that $\Vdash _{\mathbb {P}}$ “ $\mathring {X}$ is dense in $\mathring {t}^\uparrow $ ” and the proof below can be easily relativized to $\mathring {t}^\uparrow \simeq \,\!^{<\omega } 2$ ). Let $\mathring {f}$ be a $\mathbb {P}$ -name for an element of $\,\!^\omega \omega $ such that

$$ \begin{align*} \Vdash_{\mathbb{P}}\mathring{f}(n)=\max\big\{\min\big\{|t|:s\subseteq t\in \mathring{X}\big\}:s\in\,\!^n 2\big\}\;\;\text{for every}\;n.\end{align*} $$

We claim that $\mathring {f}$ is dominating over $\,\!^\omega \omega \cap V$ : Let $g\in \,\!^\omega \omega \cap V$ be strictly increasing and satisfying $g(0)>1$ . Fix an infinite maximal antichain $\{a_n:n\in \omega \}\subseteq \,\!^{<\omega } 2$ such that $|a_n|=g(n)$ , and let $A=\,\!^{<\omega } 2\setminus \bigcup \{a_n^\uparrow :n\in \omega \}\in \mathrm {Nwd}$ . It is easy to see that $|A\cap \,\!^n 2|\geq 2^{n-1}$ for every n. We know that in the extension $A\cap \mathring {X}\subseteq \,\!^{\leq N} 2$ for an $N\in \omega $ . Now if $n>N$ then we can pick a point $s\in A\cap \,\!^n 2$ such that $s\nsubseteq a_k$ for $k<n$ . As there can be no $a_k$ below s, $K=\min \{k:s\subseteq a_k\}<\omega $ , and $|a_K|>K\geq n$ . In particular, $s^\uparrow \cap \,\!^{<|a_K|}2\subseteq A$ , and hence $\mathring {f}(n)\geq \min \{|t|:s\subseteq t\in \mathring {X}\}\geq |a_K|=g(K)\geq g(n)$ .

Now we show that if $\mathbb {P}\ +$ -destroys $\mathrm {Nwd}$ , then it adds Cohen reals. Let $\mathring {X}$ be as above, then in the extension $[\mathring {X}]_\delta \ne \varnothing $ . We show that every element y of this set is Cohen over V: If $C\subseteq \,\!^\omega 2$ is a closed and nowhere dense set coded in V, then there is a tree $T\subseteq \,\!^{<\omega }2$ , $T\in \mathrm {Nwd}$ such that $C=[T]_\delta =[T]:=\{x\in \,\!^\omega 2:\forall \ n\ x\!\upharpoonright \! n\in T\}$ , in particular $|\mathring {X}\cap T|<\omega $ and hence $y\notin [T]$ .

(2-cd): This is basically [Reference Balcar, Hernández-Hernández and Hrušák1, Theorem 1.4(ii)]. Let G be $(V,\mathbb {P})$ -generic, $c\in \,\!^\omega 2\cap V[G]$ be Cohen over V, H be $(V[G],\mathring {\mathbb {Q}}[G])$ -generic, and $d\in \,\!^\omega \omega \cap V[G,H]$ be dominating over $V[G]$ . Enumerate $\,\!^{<\omega } 2=\{t_n:n\in \omega \}$ in V and for every n define $c_n\in \,\!^\omega 2\cap V[G]$ as $c_n=t_n^\frown (c(k):k\geq |t_n|)$ , then $c_n$ is also Cohen over V, and let $X=\{c_n\!\upharpoonright \! m:n\in \omega ,m\geq d(n)\}\in V[G,H]$ . Notice that X is dense in $\,\!^{<\omega } 2$ . Now if $A\in \mathrm {Nwd}\cap V$ then $|A\cap \{c_n\!\upharpoonright \! m:m\in \omega \}|<\omega $ , in particular

$$ \begin{align*} f_A(n)=\min\big\{k:\forall\;m\geq k\;c_n\!\upharpoonright\! m\notin A\big\}\end{align*} $$

is well defined, and $f_A\in \,\!^\omega \omega \cap V[G]$ . We know that $f_A(n)\leq d(n)$ for every $n\geq N_A$ for some $N_A\in \omega $ , and hence $X\cap A\subseteq \{c_n\!\upharpoonright \! m:n<N_A,m<f_A(n)\}$ .

(2-dc): If $\mathring {X}$ is a $\mathbb {P}$ -name, $p\in \mathbb {P}$ , and $p\Vdash \mathring {X}\in [\,\!^{<\omega } 2]^\omega $ , then we can assume that $\mathring {X}$ is either (i) an infinite chain, that is, $p\Vdash $ “ $\mathring {X}=\{\mathring {s}_0\subsetneq \mathring {s}_1\subsetneq \dots \}$ and $\mathring {s}_k\subseteq \mathring {x}\in \,\!^\omega 2$ for every k,” or (ii) a converging antichain, that is, there is a $\mathbb {P}$ -name $\mathring {y}$ such that $p\Vdash $ “ $\mathring {y}\in \,\!^\omega 2$ and $\forall \ n\ \forall ^\infty \ s\in \mathring {X}\ \mathring {y}\!\upharpoonright \! n\subseteq s$ .”

In the first case, as $\mathbb {P}$ satisfies the Laver-property, $\mathring {x}$ cannot be a Cohen real over V, and hence a $q\leq p$ forces that $\mathring {x}\in C$ for some nowhere dense closed set $C=[T]\in V$ , $T\in \mathrm {Nwd}\cap V$ , and so $q\Vdash |\mathring {X}\cap T|=\omega $ .

In the second case, we can shrink $\mathring {X}$ and assume that it has an enumeration $\mathring {X}=\{\mathring {s}_k:k\in \omega \}$ and there is a sequence $(\mathring {n}_k)_{k\in \omega }$ of $\mathbb {P}$ -names for an increasing sequence in $\omega $ such that p forces the following:

(♯) $$ \begin{align} \mathring{n}_0=0,\;\mathring{y}\!\upharpoonright\! \mathring{n}_k\subsetneq \mathring{s}_k,\;\mathring{y}\!\upharpoonright\! (\mathring{n}_k+1)\nsubseteq \mathring{s}_k,\;\text{and}\;|\mathring{s}_k|<\mathring{n}_{k+1}\;\text{for every }k.\end{align} $$

Now define the $\mathbb {P}$ -names $\mathring {E}_m$ as follows: p forces that if $m\in [\mathring {n}_k,\mathring {n}_{k+1})$ and $m>|\mathring {s}_k|$ then $\mathring {E}_m=\{\mathring {y}\!\upharpoonright \! m\}$ , and if $m\in [\mathring {n}_k,\mathring {n}_{k+1})$ and $m\leq |\mathring {s}_k|$ then $\mathring {E}_m=\{\mathring {y}\!\upharpoonright \! m,\mathring {s}_k\!\upharpoonright \! m\}$ . Now $\mathring {E}_m\subseteq \,\!^m 2$ is of size $\leq 2$ , hence applying the Laver property, there are a $q\leq p$ and a sequence $F_m\subseteq [\,\!^m 2]^{\leq 2}$ in V such that $|F_m|= m+1$ and $q\Vdash \mathring {E}_m\in F_m$ for every m, in particular, if $F^{\prime }_m=\bigcup F_m\in [\,\!^m 2]^{\leq 2m+2}$ then $q\Vdash \mathring {E}_m\subseteq F^{\prime }_m$ for every m. Let $A=\{t\in \,\!^{<\omega } 2:\forall \ m\leq |t|\ t\!\upharpoonright \! m\in F^{\prime }_m\}$ . Then $A\in \mathrm {Nwd}$ because for every $t\in \,\!^{<\omega } 2$ there is a $t'\supseteq t$ such that $t'\notin F^{\prime }_{|t'|}$ and hence no extension of $t'$ belongs to A, and of course $q\Vdash \mathring {X}\subseteq A$ .

We show that $\mathbb {P}\ast \mathbb {C}$ cannot $+$ -destroy $\mathrm {Nwd}$ . First notice that if $\mathring {X}$ is a $\mathbb {C}$ -name for a dense subset of $\,\!^{<\omega } 2$ , then there is a countable family $\{Y_n:n\in \omega \}$ of dense subsets of $\,\!^{<\omega }2$ such that if an $A\in \mathrm {Nwd}$ has infinite intersection with all $Y_n$ (there is always such an A because each $Y_n$ is dense) then $\Vdash _{\mathbb {C}}|A\cap \mathring {X}|=\omega $ . Why? Enumerate $\mathbb {C}=\{q_n:n\in \omega \}$ and define $Y_n=\{s\in \,\!^{<\omega }2:\exists\ q'\leq \ q_n\ q'\Vdash s\in \mathring {X}\}$ . It is easy to see that this family satisfies our requirements. Now if $\mathring {X}$ is a $\mathbb {P}\ast \mathbb {C}$ -name and $(p,q)\Vdash \mathring {X}\in \mathrm {Nwd}^+$ then we can assume that $(p,q)$ forces that $\mathring {X}$ is dense (because a condition below $(p,q)$ decides where $\mathring {X}$ is dense and we can work inside that cone in $\,\!^{<\omega }2$ ). Therefore there are $\mathbb {P}$ -names $\mathring {Y}_n$ for dense subsets of $\,\!^{<\omega }2$ such that p forces the following: “If $A\in \mathrm {Nwd}$ and $|A\cap \mathring {Y}_n|=\omega $ for every n, then $q\Vdash _{\mathbb {C}}|A\cap \mathring {X}|=\omega $ .” Working in $V^{\mathbb {P}}$ , it is trivial to construct an antichain $\mathring {Z}\subseteq \,\!^{<\omega }2$ satisfying $(\sharp )$ which has infinite intersection with all $\mathring {Y}_n$ , and hence there is an $A\in \mathrm {Nwd}\cap V$ covering $\mathring {Z}$ . It follows that $(p,q)\Vdash |A\cap \mathring {X}|=\omega $ .⊣