$n$th-power of ccc posets We know that it is relatively consistent with $\textbf{ZFC}$ that there is a ccc poset $\mathbb{P}$ such that its cartesian square $\mathbb{P} \times \mathbb{P}$ is not ccc.
Indeed, if $\mathbb{P}=T$ is a Suslin tree then $T \times T$ is not ccc.  
Is it consistent with $\textbf{ZFC}$ that there a ccc poset $\mathbb{P}$ such that $\mathbb{P} \times \mathbb{P}$ is ccc but the cube $\mathbb{P} \times \mathbb{P} \times \mathbb{P}$ is not ccc?
More generally, is it consistent with $\mathrm{ZFC}$ that there is a poset $\mathbb{P}$ such that the $n$th-power $\mathbb{P}^n$ is ccc but $\mathbb{P}^{n+1}$ is not ccc (for every $1<n<\omega$)?
 A: Yes, it is consistent.


*

*In

Fred Galvin, Chain conditions and products, Fund. Math. vol.108 (1980), pp.33-48, MR585558, link.

it is shown to hold under $\mathsf{CH}$.
The basic idea behind this proof is as follows.  For any $K \subseteq [ \omega_1 ]^2$ define $\mathbb{P}_K$ to be the family of all finite $K$-homogeneous subsets of $\omega_1$ (i.e., all finite $F \subseteq \omega_1$ such that $[F]^2 \subseteq K$).  Then $\mathbb{P}_K$ is partially ordered by $\subseteq$.  Furthermore, it can be easily shown that if $K,L$ are disjoint subseteq of $[\omega_1]^2$, then $\mathbb{P}_K \times \mathbb{P}_L$ is not ccc.
Galvin defines a subset $K \subseteq [\omega_1]^2$ to be big if whenever $H_0 , \ldots , H_{n-1} \subseteq [ \omega_1 ]^2$ are such that $K \subseteq \bigcap_{i < n} H_i$, then $\mathbb{P}_{H_0} \times \cdots \times \mathbb{P}_{H_{n-1}}$ is ccc.  $\mathsf{CH}$ implies that there are $\aleph_0$ many (actually, $\aleph_1$ many) pairwise disjoint big subsets of $[\omega_1]^2$.
Taking pairwise disjoint big $K_0 , \ldots , K_n \subseteq [ \omega_1 ]^2$, for each $i \leq n$ define $H_i = \bigcup_{j \in (n+1) \setminus \{ i \}} K_j$.  Note that for any nonempty $a \subseteq n+1$, we have that $\bigcap_{i \in a} H_i = \varnothing$ iff $a = n+1$. Then $\mathbb{P}_{H_0} , \ldots , \mathbb{P}_{H_n}$ are posets such that for any nonempty $a \subseteq n+1$ the product $\prod_{i \in a} \mathbb{P}_{H_i}$ is ccc iff $a \neq n+1$.
Then the sum $\mathbb{P} = \mathbb{P}_{H_0} \oplus \cdots \oplus \mathbb{P}_{H_n}$ has the property that $\mathbb{P}^n$ is ccc, but $\mathbb{P}^{n+1}$ is not.

*In

William G. Fleissner, Some spaces related to topological inequalities proven by the Erdős-Rado theorem, Proc. AMS vol.71 (1978), pp.313-320, MR0493930, link

it is demonstrated that this follows after adding $\omega_1$-many Cohen reals.

*In 

Stevo Todorcevic, Remarks on chain conditions in products. Compositio Math. vol.55 (1985), pp.295–302, MR0799818, link

the result is also demonstrated to hold after adding $\omega_1$-many random reals.
The last two articles actually yield stronger results: after adding $\kappa$-many Cohen or random reals (where $\kappa$ has uncountable cofinality), for each $n < \omega$ there is a poset $\mathbb{P}$ such that $\mathbb{P}^n$ is ccc, but $\mathbb{P}^{n+1}$ is not $\kappa$-cc.
