Several days ago a friend asked me the following:

We know that in $\mathcal P(\mathbb N)$ we can find a family of size continuum that every [distinct] two intersect in a finite set. Can we do that with $\mathcal P(\mathbb R)$, that is a family of size $2^{\frak c}$ many subsets of real numbers that the intersection of any [distinct] two is finite, or at least less than $\frak c$?

The question, if so, asks about $(2^{\frak c})^+$-c.c. in the Boolean algebras $\mathcal B_\kappa=\mathcal P(\mathbb R)/\sim_\kappa$ where $\sim_\kappa$ is the equivalence relation defined as $A\sim B\iff |A\triangle B|<\kappa$. The first question asks for $\mathcal B_\omega$ and the latter asks for $\mathcal B_\mathfrak c$.

Assuming GCH (or at least that $2^{\frak c}=\aleph_2$) gives a relatively simple positive answer to the latter question:

Consider the tree $2^{<\omega_1}$, it is of size $\aleph_1$ so we can encode the nodes as a real numbers. This tree has $2^{\omega_1}=\aleph_2$ many branches, each defines a subset of $\mathbb R$ using the encoding, and every distinct two branches meet at most at countable set of points.

I consulted with several other folks from the department and I was told that most of these questions are very well known, so an answer about consistency and provability is almost certainly out there. Naive Google search got me nowhere, so I came to ask here the following:

In the particular case of the question above, can we say anything in ZFC about the chain-condition of $\mathcal B_\kappa$ for $\omega\leq\kappa\leq\frak c$?

My partial answer above shows that with GCH we have an answer for $\cal B_\frak c$, but does that also answer $\cal B_\omega$ or do we need to assume stronger principles as $\lozenge$ for suitable cardinals?

How far does this generalized, when replacing $2^\omega$ by any infinite cardinal $\mu$, and asking the similar question about $(2^\mu)^+$-c.c. in the similar quotients?

I'd be glad to have a reference to a survey of such results, if it exists.

Surveys in combinatorics 1987LMS Lecture Notes 123 - ams.org/mathscinet-getitem?mr=905279 – François G. Dorais♦ Jun 8 '12 at 14:18