This question is similar to [a question I asked last year](https://mathoverflow.net/questions/402333), but I'm not asking for the same thing.




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Let $S$ be a set of ordinals, and consider the Levy collapse $\operatorname{Col}(\omega,S)$. Let $G$ be $\operatorname{Col}(\omega,S)$-generic. Must every unbounded subset of $\omega$ in $V[G]$ contain an unbounded further subset in the ground model $V$? I believe the answer is yes, with a proof similar to the linked question, but I am unsure (the proof cannot be brought over verbatim).


I am also interested in the more general question: Is it possible to characterise posets $\mathbb{P}$ such that for all filters $G$ which are $\mathbb{P}$-generic, every subset of $\omega$ in $V[G]$ has an unbounded further subset in $V$?



Unbounded subset of $\omega$ in $V[G]$ has an unbounded subset in $V$?
No, this does not hold.


Given an infinite set $A$, let $\hat{A}=\{\langle n, a\_1,...,a\_n\rangle: n\in\mathbb{N}\}$ where $\langle\rangle$ is an appropriate "tupling" operation and $a\_i$ is the $i$th smallest element of $A$. Note that any infinite subset of $\hat{A}$ lets us reconstruct all of $\hat{A}$ itself. This means in particular that if $X\in\mathcal{P}^{V[G]}(\omega)\setminus\mathcal{P}^V(\omega)$ then $\hat{X}$ is an infinite set with no infinite subset in $V$. So indeed **any** forcing which adds a real - including $Col(\omega,S)$ whenever $S$ is uncountable - will add infinite subsets of $\omega$ not containing any infinite subset from the ground model.


In fact, an even stronger phenomenon can occur. Identify an (infinite) set of natural numbers $A$ with its **principal function** $p\_A$ which on input $i$ gives the $i$th smallest element of $A$. Note that if $A\subseteq B$ then $p\_A$ grows at least as fast as $p\_B$. Now $Col(\omega,2^{\aleph\_0})$ adds a function dominating all ground model functions, which is a massive strengthening of "adding a real with no infinite ground model subset." This raises an interesting follow-up question: must $Col(\omega,\omega\_1)$ add a function not escaped by any function in the ground model? This takes us into the realm of **cardinal chracteristics of the continuum**, and see in particular [this old question of mine](https://mathoverflow.net/questions/77891/can-we-collapse-omega-1-to-omega-without-adding-a-dominating-real).


