Suppose $\mathbb{P}$ is a separative partial order of uniform density $\kappa$, i.e. for all $p \in \mathbb{P}$, the least size of dense set below $p$ is $\kappa$. Does forcing with $\mathbb{P}$ add an unbounded $A \subseteq \kappa$ such that for all unbounded $X \subseteq \kappa$ in the ground model, $X \cap A \not= \emptyset$ and $X \setminus A \not= \emptyset$?

In my answer here, I showed one half of this: There is an unbounded $A$ such that $X \setminus A \not= \emptyset$ for all ground model unbounded $X$. However, the method of construction does not yield the stronger "splitting" property.

Answers with additional assumptions such as $\kappa$ is regular and/or $\mathbb{P}$ is $\kappa$-c.c. would still be useful. But we already know the case $\kappa = \omega$ is true!

Suppose $\mathbb{P}$ is a separative partial order of uniform density $\kappa$, i.e. for all $p \in \mathbb{P}$, the least size of dense set below $p$ is $\kappa$. Does forcing with $\mathbb{P}$ add an unbounded $A \subseteq \kappa$ such that for all unbounded $X \subseteq \kappa$ in the ground model, $X \cap A \not= \emptyset$ and $X \setminus A \not= \emptyset$?

In my answer here, I showed one half of this: There is an unbounded $A$ such that $X \setminus A \not= \emptyset$ for all ground model unbounded $X$. However, the method of construction does not yield the stronger "splitting" property.

Answers with additional assumptions such as $\kappa$ is regular and/or $\mathbb{P}$ is $\kappa$-c.c. would still be useful. But we already know the case $\kappa = \omega$ is true!

Suppose $\mathbb{P}$ is a partial order of uniform density $\kappa$, i.e. for all $p \in \mathbb{P}$, the least size of dense set below $p$ is $\kappa$. Does forcing with $\mathbb{P}$ add an unbounded $A \subseteq \kappa$ such that for all unbounded $X \subseteq \kappa$ in the ground model, $X \cap A \not= \emptyset$ and $X \setminus A \not= \emptyset$?

In my answer here, I showed one half of this: There is an unbounded $A$ such that $X \setminus A \not= \emptyset$ for all ground model unbounded $X$. However, the method of construction does not yield the stronger "splitting" property.

Answers with additional assumptions such as $\kappa$ is regular and/or $\mathbb{P}$ is $\kappa$-c.c. would still be useful. But we already know the case $\kappa = \omega$ is true!

Suppose $\mathbb{P}$ is a separative partial order of uniform density $\kappa$, i.e. for all $p \in \mathbb{P}$, the least size of dense set below $p$ is $\kappa$. Does forcing with $\mathbb{P}$ add an unbounded $A \subseteq \kappa$ such that for all unbounded $X \subseteq \kappa$ in the ground model, $X \cap A \not= \emptyset$ and $X \setminus A \not= \emptyset$?

In my answer here, I showed one half of this: There is an unbounded $A$ such that $X \setminus A \not= \emptyset$ for all ground model unbounded $X$. However, the method of construction does not yield the stronger "splitting" property.

Answers with additional assumptions such as $\kappa$ is regular and/or $\mathbb{P}$ is $\kappa$-c.c. would still be useful. But we already know the case $\kappa = \omega$ is true!