Proof. Let $D\subseteq \mathbb {B}$ be dense. For $d\in D$ , let

$$\begin{align*}u_d:=\bigwedge\{a\in \mathbb{A}:d\le a\}. \end{align*}$$

We will be done if we can argue that $\{u_d:d\in D\}$ is dense in $\mathbb {A}$ . So, fix $a\in \mathbb {A}$ . By the density of $D$ , there exists some $d\in D$ with $d\le a$ , hence $u_d\le a$ .

Proof. For the homogeneity, see [Reference Koppelberg8, Corollary 12.5]. The uniform density assertion follows from homogeneity and the fact that $d(\mathbb {C}_\kappa )$ is also the smallest size of a dense subset of $\mathrm {Add}(\omega ,\kappa )$ , which is easily seen to be $\kappa $ .

Proof. Suppose not. Then we can find $G$ which is $\mathbb {C}_\kappa $ -generic over $V$ and $H$ which is $\mathbb {C}_\lambda $ -generic over $V$ such that $H\in V[G]$ . Obviously, $V\subseteq V[H]\subseteq V[G]$ . By the intermediate model theorem (see [Reference Jech7, Lemma 15.43]), there exists $\mathbb {B}$ a complete regular subalgebra of $\mathbb {C}_\kappa $ such that $V[H]=V[G\cap \mathbb {B}]$ . By Lemma 2.3, there exist $b\in \mathbb {B}$ and $c\in \mathbb {C}_\lambda $ such that $\mathbb {B}_b \cong (\mathbb {C}_\lambda )_c$ (in $V$ ). But then, by Lemmas 2.2 and 2.4,

$$\begin{align*}\lambda = d((\mathbb{C}_\lambda)_c)=d(\mathbb{B}_b)\le d(\mathbb{B})=\kappa, \end{align*}$$

which is a contradiction.

Proof. Each part is a consequence of the ccc. For the first part let $\{f_\alpha : \alpha \in \omega _1\}$ be the ground model Borel functions and let $A = \{a_\alpha : \alpha \in \aleph _\omega \}$ be the set of Cohen generic reals coming from $G_0$ . If $\dot {x}$ is a nice name for a real then it is well known, see, e.g., [Reference Zapletal12, Theorem 4.1.2], that there are $c \in [\aleph _\omega ]^\omega \cap V$ so that $\dot {x}$ is forced to be in $V[a_\xi : \xi \in c]$ and in that model there is an $\alpha < \omega _1$ so that $\dot {x}^{G_0} = f_\alpha (\{a_\xi : \xi \in c\})$ . For the second part, observe that any $\aleph _\omega $ -many reals and any $\aleph _1$ -many Borel functions were added by $\aleph _\omega $ -many of the Cohen reals in $G_1$ and hence there is a Cohen real which is generic over an intermediate model containing all of them.

Proof. For (A) note that if $\theta = 2^\sigma $ then letting $\mathcal F_c := \mathcal {P}(c)$ satisfies the requirements of the principle.

For (B), note that since $\sigma \leq \theta $ we have that for any $\overline {\mathcal {F}}$ as in the statement of Definition 1.1 we can define $\overline {\mathcal {F}}'$ by $\mathcal {F}^{\prime }_c := \{\emptyset \}\cup \{\{\alpha \}\mid \alpha \in c\} \cup \mathcal {F}_c$ and $\mathcal {F}^{\prime }_c$ will still have size at most $\theta $ . Now for each $A \subseteq \lambda $ of size ${<}\lambda $ , by $\lambda =\mu ^+$ we can enumerate $A$ (possibly with repetitions) as $A=\{\alpha _\xi :\xi <\mu \}$ . Then $\{\alpha _\xi \}\cap c \in \mathcal {F}^{\prime }_c$ for any $c\in [\lambda ]^\sigma $ .

For (C), any witness to $\Pr (\sigma ,\theta _0,\mu _0,\lambda )$ will witness $\Pr (\sigma ,\theta _1,\mu _0,\lambda )$ . For the monotonicity in $\mu $ , let $\mathcal {F}$ witness $\Pr (\sigma ,\theta _0,\mu _0,\lambda )$ and note that we may assume without loss of generality that $\emptyset \in \mathcal {F}_c$ for every $c\in [\lambda ]^{\sigma }$ . Given $A\in [\lambda ]^\lambda $ , let ${\langle {B_\xi :\xi <\mu _0}\rangle }$ witness Definition 1.1(C) and set $B_\xi =\emptyset $ for $\xi \in [\mu _0,\mu _1)$ . This works.

Proof. Fix $\sigma , \theta , \mu , \lambda ,$ and $\mathbb P$ as in the statement of the theorem. Since $\mathbb P$ has the $\sigma ^+$ -covering property, there is a name $\dot \Phi $ such that the following is forced by $\mathbb P$ : “ $\dot \Phi :[\lambda ]^{|\sigma |}\to ([\lambda ]^\sigma )^V$ and for all $c\in [\lambda ]^{|\sigma |}$ , $c\subseteq {\dot \Phi }(c)$ .”

Working in $V^{\mathbb P}$ , for each $c \in [\lambda ]^{|\sigma |}$ let $\mathcal {F}_c = \{a\cap c : a \in V \cap \mathcal {P}(\dot {\Phi }(c))\}$ . Note that for every $c$ we have that $\mathcal {F}_c$ has cardinality at most $(2^\sigma )^V = |\theta |$ and hence $\overline {\mathcal {F}} = \langle \mathcal {F}_c : c \in [\lambda ]^\sigma \rangle $ satisfies the first two clauses of Definition 1.1. Back in $V$ let $\dot {\overline {\mathcal {F}}}$ name this sequence and for each $\dot {c}$ let $\dot {\mathcal {F}}_{\dot {c}}$ be the name for the associated element. We need to see that the third clause of Definition 1.1 is forced.

Suppose that $\dot {A}$ is a $\mathbb P$ -name such that $\Vdash \dot A\subseteq \lambda $ . For each $q\in \mathbb P$ , let $B_q=\{\xi <\lambda :q\Vdash \xi \in \dot {A}\}$ . Note that $\Vdash \dot {A}=\bigcup _{q\in \dot {G}}B_q$ . Now let $\dot c$ be a $\mathbb P$ -name for a $|\sigma |$ -sized subset of $\lambda $ . If $q\in \mathbb P$ , then $\Vdash B_q\cap \dot {c}=B_q\cap \dot {\Phi }(\dot c)\cap \dot {c}\in \dot {\mathcal {F}}_{\dot c}$ because $\Vdash \dot {\Phi }(\dot c)\in V$ .

Proof. We argue by contradiction. By homogeneity of $\mathbb P$ , we may assume that $\Vdash _{\mathbb P} \Pr (\sigma ,\theta ,\mu ,\lambda )$ . Let $\dot {\bar {\mathcal {F}}}$ witness this. We aim to produce a name for a $\lambda $ -sized subset of $\lambda $ which cannot be decomposed as in Definition 1.1. Actually, we will produce $\theta ^+$ many sets and show by contradiction they cannot all be decomposed as we want.

Let $\dot g$ be a $\mathbb P$ -name for the generic function $\lambda \to 2$ . For $\xi < \theta ^+$ , let $\dot {A}_\xi $ be a name for the set $\{\alpha <\lambda : \dot {g}(\theta ^+\cdot \alpha + \xi )= 1\}$ . An easy genericity argument confirms that each $\dot {A}_\xi $ is forced to have has size $\lambda $ and that they are all distinct.

Since we assume that it is forced that $\mathsf {Pr}(\sigma , \theta , \mu , \lambda )$ holds we have names $\dot {B}_{\xi , i}$ for $\xi < \theta ^+$ and $i < \mu $ so that

$$\begin{align*}\Vdash \dot{A}_\xi = \bigcup \{\dot{B}_{\xi, i} : i \in \mu\}. \end{align*}$$

Definition 1.1(C) holds. Our goal is to find a $c \in [\lambda ]^\sigma $ such that $c\cap B_{\xi , i}$ are pairwise distinct for $\theta ^+$ many pairs $(\xi , i)$ . Since each of these must, by definition, be in $\dot {\overline {\mathcal {F}}}_c$ , we will contradict the assumption that this set has size at most $\theta $ .

To find this set of pairs $(\xi , i)$ we will repeatedly thin out the sets we have constructed to get more and more homogeneity. To begin this process, for each $\xi < \theta ^+$ and $\alpha < \lambda $ we can pick $(p_{\xi , \alpha }, i_{\xi , \alpha })$ so that $i_{\xi ,\alpha } < \mu $ and $p_{\xi , \alpha } \Vdash \check {\alpha } \in B_{\xi , i_{\xi , \alpha }}$ . Note that any such $p_{\xi , \alpha }$ forces in particular that $\check {\alpha } \in \dot {A}_\xi $ and therefore we have that

(1) $$ \begin{align} \langle \theta^+\cdot\alpha + \xi, 1 \rangle \in p_{\xi, \alpha}. \end{align} $$

Applying the pigeonhole principle judiciously to this situation and further thinning out we can find moreover for each $\alpha < \lambda $ a set $u_\alpha \in [\theta ^+]^{\theta ^+}$ so that:

• • the sequence $\vec {i}_\alpha = {\langle {i_{\xi , \alpha } : \xi \in u_\alpha }\rangle }$ is bounded in $\mu $ ;

• • $\langle \mathrm {dom}(p_{\xi , \alpha }) : \xi \in u_\alpha \rangle $ forms a $\Delta $ -system;

• • $\langle p_{\xi ,\alpha }:\xi \in u_\alpha \rangle $ are pairwise compatible with common intersection $r_\alpha $ .

Fixing $\alpha $ , note that at most finitely many $\xi \in u_\alpha $ can have $\theta ^+\cdot \alpha + \xi \in {\mathrm {dom}}(r_\alpha )$ . By throwing out these finitely many elements we can assume that

(2) $$ \begin{align} \forall \alpha<\lambda \forall \xi \in u_\alpha\, [\theta^+\cdot\alpha+\xi\not\in{\mathrm{dom}}(r_\alpha)]. \end{align} $$

Next we aim for even more homogeneity. Let $S:=\{\alpha \in [\mu ,\lambda ):\mathrm {cf}(\alpha )=\theta ^{++}\}$ . For each $\delta \in S$ , set $\beta _\delta :=\min (X_*\setminus \delta )$ .

Since $\delta \mapsto \gamma _\delta $ is a regressive function on $S$ we can find a stationary set $S_* \subseteq S$ with the property that for all $\delta \in S_*$ we have $\gamma _\delta = \gamma _*$ for some fixed $\gamma _*$ . Note that $|\gamma _*| = \mu $ . Now for each $\alpha \in S_*$ consider the function $\alpha \mapsto {\sup }[\bigcup \{{\mathrm {dom}}(p_{\xi , \beta _\alpha }) : \xi \in u_*\}]$ . We can find a club $E$ bounding this function—i.e., for every $\alpha \in S_*$ we have

(3) $$ \begin{align} {\sup}\left[\bigcup \{{\mathrm{dom}}(p_{\xi, \beta_\alpha}) : \xi \in u_*\}\right] < \mathrm{min}[ E \setminus (\alpha + 1)]. \end{align} $$

Let $X_{**}$ be the set $\{\beta _\delta : \delta \in S_* \cap E\}$ .

Claim 4.4. Let $\delta ,\varepsilon \in S_\ast \cap E$ with $\delta <\varepsilon $ . If $\beta _\delta ,\beta _\varepsilon $ are distinct elements of $X_{**}$ then we have that

$$\begin{align*}\bigcup \{{\mathrm{dom}}(p_{\xi, \beta_\delta}) : \xi \in u_{*}\} \cap \bigcup \{{\mathrm{dom}}(p_{\xi, \beta_\varepsilon}) : \xi \in u_{*}\} \subseteq \gamma^*.\end{align*}$$

Proof. Fix $\xi ,\eta \in u_\ast $ . By (3), $\mathrm {dom}(p_{\xi \beta _\delta })\subseteq \varepsilon $ , hence

$$ \begin{align*} \mathrm{dom}(p_{\xi,\beta_\delta})\cap{\mathrm{dom}}(p_{\eta,\beta_\varepsilon})& ={\mathrm{dom}}(p_{\xi,\beta_\delta})\cap \mathrm{dom}(p_{\eta,\beta_\varepsilon})\cap \varepsilon \\ & = \mathrm{dom}(p_{\xi,\beta_\delta})\cap \mathrm{dom}(p_{\xi,\beta_\varepsilon}) \cap \gamma_*,\\ & \subseteq \gamma^*, \end{align*} $$

where the second equality is by the choice of $\gamma _*$ .

Let $\langle W_n : n < \omega \rangle $ be a $\subseteq $ -increasing sequence of sets of size ${<}\mu $ so that $\bigcup _{n < \omega } W_n = \gamma ^*$ . For $\delta \in E \cap S_*$ and $\xi \in u_*$ there is an $n_{\delta , \xi } \in \omega $ so that

(4) $$ \begin{align} {\mathrm{dom}}(p_{\xi, \beta_\delta}) \cap \gamma_* \subseteq W_{n_{\delta, \xi}} \end{align} $$

simply by the finiteness of the domains. But this implies that for every $\delta \in S_* \cap E$ there is a $u_{*, \delta } \in [u_*]^{\theta ^+}$ so that $n_{\delta , \xi } = n_\delta $ for some fixed $n_\delta \in \omega $ and every $\xi \in u_{*, \delta }$ . By a pigeonhole argument, we can find a stationary set $S_{**} \subseteq S_{*} \cap E$ , together with $n\in \omega $ , $u_{**}\in [u_*]^{\theta ^+}$ and a family ${\langle {p_\xi :\xi \in u_{**}}\rangle }$ so that, for every $\delta \in S_{**}$ , the following hold:

• • $n_\delta = n$ ;

• • $u_{*, \delta } = u_{**}$ ;

• • ${\langle {p_{\xi , \beta _\delta } \upharpoonright W_n : \xi \in u_{**}}\rangle } = {\langle {p_\xi : \xi \in u_{**}}\rangle }$ .

Reaching back to the beginning of the proof we can observe that ${\langle {p_\xi : \xi \in u_{**}}\rangle }$ forms a $\Delta $ -system by merit of being a family of restrictions of $\langle p_{\xi , \alpha } : \xi \in u_*\rangle $ which was constructed to be a $\Delta $ -system. Therefore, we can find a finite set $y \subseteq W_n$ and $p^*$ so that for distinct $\xi _1, \xi _2 \in u_{**}$ we have ${\mathrm {dom}}(p_{\xi _1}) \cap {\mathrm {dom}}(p_{\xi _2}) = y$ and $p_{\xi _1} \upharpoonright y = p_{\xi _2} \upharpoonright y = p^*$ . Let $c \subseteq \{\beta _\delta : \delta \in S_{**}\}$ be of size $\sigma $ . Let $\dot {x}_\xi $ be a name for $\dot {B}_{\xi , i_\xi } \cap \check {c}$ . By the assumption on the $\dot {B}_{\xi , i}$ ’s and the formulation of Definition 1.1 we must have that $\Vdash \dot {x}_\xi \in \dot {\overline {\mathcal {F}}}_{\check {c}}$ for every $\xi \in u_{**}$ .

We will now contradict (B) by showing that the following claim holds.

Proof of claim

Let $\dot {u}_{\dagger }$ be the name for the set of $\xi \in u_{**}$ so that $p_{\xi } \in \dot {G}$ . This set is forced by $p^*$ to have size $\theta ^+$ . To see this, observe that otherwise some $q \leq p^*$ forces it to have size at most $\theta $ so fix such a $q$ and let $\delta < \theta ^+$ be an ordinal so that $q \Vdash \dot {u}_{\dagger } \subseteq u_{**} \cap \delta $ . Fix an $\xi \in u_{**}$ . Since $q$ strengthens $p^*$ , if $p_\xi $ and $q$ are incompatible, there is an $\eta \notin y$ so that $p_\xi (\eta )\neq q(\eta )$ and in particular their domains intersect outside of $y$ . Since the domains of the $p_\xi $ ’s however form a $\Delta $ -system there is at most one such $\xi $ for $\eta \in \mathrm {dom}(q) \setminus y$ . As $q$ has finite domain, there are cofinitely many $\xi \in u_{**}$ so that $p_\xi $ and $q$ are compatible, so in particular there is one such $\xi $ with $\xi>\delta $ . But now if $r \leq p_\xi , q$ then $r \Vdash \xi \in \dot {u}_{\dagger } \setminus \delta $ , which is absurd.

Now, towards a contradiction suppose that $\xi _1 \neq \xi _2 \in u_{**}$ and let $p' \leq p^*$ force that $\xi _1, \xi _2 \in \dot {u}_{\dagger }$ but $\dot x_{\xi _1} = \dot {x}_{\xi _2}$ . By Claim 4.4, for each $\zeta \in \mathrm {dom}(p')\setminus \gamma _\ast $ , there exists at most one $\beta \in c$ such that $\zeta \in \mathrm {dom}(p_{\xi _1,\beta })$ . Since $c$ is infinite and $\mathrm {dom}(p')$ is finite, it follows that we can find some $\beta \in c$ such that

(5) $$ \begin{align} \forall \zeta\in\mathrm{dom}(p')\setminus \gamma_\ast [\zeta\not\in\mathrm{dom}(p_{\xi_1,\beta})] \end{align} $$

and

(6) $$ \begin{align} \theta^+\cdot\beta+\xi_2\not\in\mathrm{dom}(p'). \end{align} $$

By (1), recall that we have that $\theta ^+\cdot \beta +\xi _2\in \mathrm {dom}(p_{\xi _2,\beta })$ . On the other hand, by (2),

$$\begin{align*}\theta^+\cdot\beta +\xi_2\not\in \mathrm{dom}(r_{\beta})=\mathrm{dom}(p_{\xi_1\beta})\cap \mathrm{dom}(p_{\xi_2\beta}), \end{align*}$$

and therefore $\theta ^+\cdot \beta +\xi _2\not \in \mathrm {dom}(p_{\xi _1\beta })$ . Define

$$\begin{align*}r:= p' \cup [p_{\xi_1, \beta} \mathord{\upharpoonright} ({\mathrm{dom}}(p_{\xi_1, \beta}) \setminus {\mathrm{dom}}(p^*))] \cup \{\langle \theta^+\cdot \beta + \xi_2, 0\rangle \}. \end{align*}$$

We need to argue that $r$ is a condition, the issue being the compatibility of $p'$ and $p_{\xi _1,\beta }$ . Fix $\zeta \in \mathrm {dom}(p')\cap \mathrm {dom}(p_{\xi _1,\beta })$ . By (5), $\zeta <\gamma ^*$ , hence $\zeta \in W_n$ by (4). But $p_{\xi _1,\beta }\mathord {\upharpoonright } W_n=p_{\xi _1}$ and $p'\Vdash \xi _1\in u_{\dagger }$ , so $p'$ and $p_{\xi _1}$ are compatible, therefore $p_{\xi _1,\beta }(\zeta )=p'(\zeta )$ , as desired.

We have shown that $r$ is a condition extending $p_{\xi _1, \beta }$ , so $r\Vdash \beta \in B_{\xi _1,i_{\xi _1}}$ by the choice of $p_{\xi _1,\beta }$ . On the other hand, $r\Vdash \beta \notin \dot A_{\xi _2}$ because $\langle \theta ^+\cdot \beta + \xi _2, 0\rangle $ , and therefore $r\Vdash \beta \notin \dot {B}_{\xi _2, i}$ for any $i < \mu $ . This contradicts the assumption that $p'$ forces $\dot x_{\xi _1} = \dot x_{\xi _2}$ , thus proving the claim.

By the last claim, $p^*$ forces that $\dot {\overline {\mathcal {F}}}_{\check {c}}$ has size at least $\theta ^+$ , so we have reached our final, promised, contradiction so we are done.

Question 4.7. What applications does $\mathsf {Pr}(\sigma , \theta , \mu , \lambda )$ have to other types of structures? Does some mathematical consequence of $\mathsf {Pr}(\sigma , \theta , \lambda , \mu )$ , for example, some statement about free groups along the lines of what is found in [Reference Shelah, Boffa, van Dalen and McAloon10], serve to distinguish adding $\aleph _\omega $ from adding $\aleph _{\omega +1}$ -many Cohen reals?

Question 4.8. What consequences more generally does $\mathsf {Pr} (\sigma , \theta , \mu , \lambda )$ have for the choices of $\sigma $ , $\theta $ , $\mu ,$ and $\lambda $ considered in this article? For example, could it be equivalent to some partition principle or its negation?

Question 4.9. Is it consistent that $\Pr (\sigma ,\theta ,\mu ,\lambda _0) \wedge \neg \Pr (\sigma ,\theta ,\mu ,\lambda _1)$ holds for some $\lambda _0<\lambda _1$ ?

Question 4.10. Can $\Pr $ exhibit any incompactness? For example, is it consistent that $\Pr (\aleph _0,\aleph _1,\aleph _2,\aleph _n)$ holds for every $n>2$ but $\Pr (\aleph _0,\aleph _1,\aleph _2,\aleph _\omega )$ fails?