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 $\mathcal B\_\mathfrak c$.




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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.




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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:


1. In the particular case of the question above, can we say anything in ZFC about the chain-condition of $\mathcal B\_\kappa$ for $\omega\leq\kappa\leq\frak c$?
2. 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?
3. 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.



Chain conditions in quotients of power sets
The answer to your first question (with finite intersections) is negative.


Indeed, if $X$ is an infinite set and $I$ has cardinality greater than that of $X^{\aleph\_0}$ then $X$ can't contain $I$ distinct subsets with pairwise finite intersection. This answers your question since $c^{\aleph\_0}=c$.


Indeed, let $(A\_i)\_{i\in I}$ be a family of subsets of $X$ with pairwise finite intersection. Let $B\_i$ be the set of infinite countable subsets of $X$ contained in $A\_i$. Then the $B\_i$ are pairwise disjoint. Moreover, $B\_i$ is empty only when $A\_i$ is finite, and we can remove such exceptional $i$'s because the number of finite subsets of $X$ is only the cardinality of $X$. 


The $B\_i$ live in the set of infinite countable subsets of $X$, which has cardinality $X^{\aleph\_0}$. So $I$ is at most the cardinal of $X^{\aleph\_0}$.


**Edit**: the obvious generalization of the argument is the following: if $\alpha,\beta,\gamma$ are infinite cardinals, and if $\alpha$ admits $\beta$ subsets with pairwise intersection of cardinal $<\gamma$, then $\beta\le\alpha^\gamma$. In particular, if $\alpha=2^\delta$ and $\gamma\le\delta$ then $\alpha^\gamma=\alpha$, so the conclusion reads as: $2^\delta$ does not admit more that $2^\delta$ subsets with pairwise intersection of cardinal $<\delta$.


