Chain conditions in quotients of power sets

Question

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.

Answer 1

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 $<\delta$.

Answer 2

It is consistent with ZFC that a set of size $\aleph_1$ does not have $2^{\aleph_1}$ subsets, each of size $\aleph_1$, with all pairwise intersections countable. This is an old result of Jim Baumgartner; see "Almost-disjoint sets, the dense-set problem, and the partition calculus", Ann. Math. Logic 9(1976), 401-439, particularly Theorem 5.6(d) and the remark on page 422 after it. [Caution: I can't check the paper itself now; I'm going by an old e-mail from Jim.]

Answer 3

I'll add in that Shelah has used pcf theory to investigate related questions. Typically these results are tucked away inside long papers dealing with other questions, but I know that the last section of [Sh:410] explicitly deals with ``strongly almost disjoint families", and characterizes their existence in terms of pcf.

For example, if $\aleph_0<\kappa\leq\kappa^{\aleph_0}<\lambda$, then the existence of a family of $\lambda^+$ sets in $[\lambda]^{\kappa}$ with pairwise finite intersection is equivalent to a ``pcf statement''.

I'm not sure which version of the paper to link to, as the published version has been reworked a few times. I THINK that the most recent version is here:

Sh:410