Suppose $\kappa$ is a regular cardinal and $P$ is a $\kappa$-c.c. partial order. I want to know when are small sets added by subforcings of size $<\kappa$. The following seems well-known:

Fact:If $\kappa$ is weakly compact, and $P$ has size $\kappa$ and is $\kappa$-c.c., then for any $P$-name $\tau$ for a set of ordinals of size $<\kappa$, there is a $Q \lhd P$ and a $Q$-name $\sigma$ such that $|Q| < \kappa$, and $1 \Vdash_P \sigma = \tau$.

I think the easiest way to see the fact is to use the extension property. Every such $P$ can be coded as a set of ordinals $A \subseteq \kappa$, and there is some transitive $X$ of rank > $\kappa$ and a set $B$ such that $(V_\kappa,\in,A) \prec (X,\in,B)$. Any $P$-name for a small set is seen by the larger structure as captured by $P$, so this reflects.

**Question 1:** Are there counterexamples for some large cardinals which are weaker than weakly compact?

If $\kappa$ is supercompact, then the same conclusion holds for all $\kappa$-c.c. $P$. This is easy to see by taking a supercompactness embedding $j$ with closure at least $|P|$ and noting that $j[P]$ is a regular suborder of $j(P)$. Supercompactness must be a total overkill hypothesis.

**Question 2:** For what cardinals $\kappa$ do we have that every $\kappa$-c.c. partial order captures small sets in small factors?

**Update:** I think I have a partial answer to Question 2. Supercompactness is indeed overkill, and weak compactness is enough after all. Let $\kappa$ be weakly compact, and $P$ be $\kappa$-c.c. Let $\theta > \kappa$ be regular such that $P \in H_\theta$. Let $\tau$ be any $P$-name for a $<\kappa$ sized set of ordinals. Let $M \prec H_\theta$ be such that $P,\tau \in M$, $|M| \leq \kappa$, and $M^{<\kappa} \subseteq M$. This is possible just because $\kappa$ is inaccessible. Then $P \cap M$ is a regular suborder of $P$, since *all* antichains contained in $P \cap M$ are members of $M$, and $M$ knows which ones are maximal. By the **Fact**, there is $Q \lhd P \cap M \lhd P$ and a $Q$-name $\sigma$ such that $|Q| < \kappa$ and $\Vdash_{P \cap M} \tau = \sigma$.

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