Preservation of chain conditions

Question

Suppose $\kappa<\lambda$ are regular cardinals, $P$ is a $\kappa$-c.c. poset, and $Q$ is a $\lambda$-c.c. poset. Does forcing with $P$ preserve the $\lambda$-c.c. of $Q$?

Answer 1

Not in general. In the paper Remarks on cellularity in products, Todorcevic shows that after adding a Cohen (or random) real, there is a ccc forcing $\mathbb P$ so that $\mathbb P\times\mathbb P$ is not $2^\omega$-cc (apparently this generalises earlier work by Fleissner and Roitman). In particular, if one blows up the contiuum to some large regular $\lambda$ by adding Cohen reals then there is a ccc $\mathbb P$ so that after forcing with $\mathbb P$, $\mathbb Q=\mathbb P$ is not $\lambda$-cc.

There are some restrictions though. Using the special case $$(2^\kappa)^+\rightarrow (\kappa^+)^2_2$$ of the Erdős-Rado theorem one can show that if $\mathbb P$ and $\mathbb Q$ are $\kappa^+$-cc then $\mathbb P\times\mathbb Q$ is $(2^\kappa)^+$-cc (similar to how one shows that weakly compact cardinals $\mu$ are productive using the partition relation $\mu\rightarrow(\mu)^2_2$). Thus under $\mathrm{GCH}$ there are no counterexample with $\mathbb P=\mathbb Q$ and $\kappa$ a successor cardinal (note that this is the case above): If $\mathbb P$ is $\eta^+=\kappa$-cc then $\mathbb P\times\mathbb P$ is $(2^ \eta)^+=\kappa^+\leq\lambda$-cc.

This also gives a bound on how badly $\lambda$-cc of $\mathbb Q$ can fail in this context: After any $\lambda$-cc forcing $\mathbb Q$ has at least the $(2^\lambda)^+$-cc (and even $(2^\eta)^+$-cc if $\lambda$=$\eta^+$).

It would be interesting to know whether the answer to your question is yes under $\mathrm{GCH}$.