# Families with finite intersection property on $\kappa>\omega$

Let $$\kappa>\omega$$ be a cardinal. We say that $${\cal A}\subseteq{\cal P}(\kappa)$$ has the finite intersection property (FIP) if $$|A|=\kappa$$ for $$A\in{\cal A}$$, and $$|A\cap B|<\aleph_0$$ for $$A\neq B\in{\cal A}$$.

For which cardinals $$\kappa>\omega$$ is there a family with FIP $${\cal A}\subseteq {\cal P}(\kappa)$$ such that $$|{\cal A}|>\kappa$$?

• Did you mean to write $|A\cap B|<\aleph_0$ for $A\neq B$? AFAIK, this is not the usual definition. Commented Oct 16, 2020 at 8:43
• I did not mean to write $|A\cap B|<\kappa$ if that is what you had in mind? Maybe I have to change my terms from "almost disjoint" to "finite intersection" families. Will edit - thanks for your comment @HannesJakob Commented Oct 16, 2020 at 9:02
• “Almost disjoint” is correct (though it is used with several variant meanings). “Finite intersection property” is something completely different: a family of sets has the FIP if every finite subfamily has nonempty intersection. Commented Oct 16, 2020 at 9:40
• I’m not sure why bof removed their comment, but I believe it is correct: any such family has cardinality at most $\kappa^\omega$, hence $\kappa^\omega>\kappa$ is a necessary condition. To see this, pick a countable infinite subset $A'\subseteq A$ in each $A\in\mathcal A$; since $A\cap B$ is finite for $A\ne B$, we must have $A'\ne B'$, thus $|\mathcal A|$ is at most the number of countable subsets of $\kappa$, i.e., $\kappa^\omega$. Commented Oct 16, 2020 at 11:02
• "finite intersection property" is not an ideal name, as it is also used to describe the property that all finite intersections are non-empty (e.g. in the definition of compactness, when using families of closed sets with the FIP instead of open coverings). Commented Oct 17, 2020 at 8:29

This is a question that connects to many things in set theory (and they are sometimes called strongly almost disjoint families").

First, an old result of Baumgartner (see Section 6 of [1]) shows that you can start with a model of GCH and force the existence of such families for a given $$\kappa$$ without collapsing cardinals or changing cofinalities. Baumgartner's forcing will result in a model in which $$\kappa\leq 2^{\aleph_0}$$. He obtains more general results: Assume GCH. Given infinite cardinals $$\nu<\kappa<\lambda$$ with $$\nu$$ regular, you can force the existence of a family $$\mathcal{A}\subseteq [\kappa]^{\kappa}$$ of size $$\lambda$$ such that any two members of $$\mathcal{A}$$ have intersection of cardinality less than $$\nu$$.

On the other hand, ZFC tells us that there are no such $$\kappa$$ above $$\beth_\omega$$, the first strong limit singular cardinal (best viewed as the supremum of the sequence $$2^{\aleph_0}, 2^{2^{\aleph_0}}, 2^{2^{2^{\aleph_0}}},..$$).

If we let $$\mu=\beth_\omega$$, then by Shelah's Revised GCH Theorem [3], for any $$\kappa>\mu$$ we can find $$\mathcal{P}\subseteq [\kappa]^{<\mu}$$ and a regular $$\sigma<\mu$$ such that

• $$|\mathcal{P}|=\kappa$$, and
• every $$A\in [\kappa]^{<\mu}$$ is a union of fewer than $$\sigma$$ members of $$\mathcal{P}$$.

(See in particular Conclusion 1.2 (4) from [3].)

Since $$\sigma$$ is regular, each subset of $$\kappa$$ of cardinality at least $$\sigma$$ must have a subset of cardinality at least $$\sigma$$ lying in $$\mathcal{P}$$. Thus, given $$\mathcal{A}\subseteq[\kappa]^\kappa$$ of cardinality $$\kappa^+$$, we can find $$X\subseteq \kappa$$ of cardinality $$\sigma$$ such that $$|\{A\in\mathcal{A}: X\subseteq A\}|=\kappa^+,$$ which is much stronger than you require.

Now what happens for $$\kappa$$ between $$2^{\aleph_0}$$ and $$\beth_\omega$$? This turns out to be connected to pcf theory, and is very much open.

For example, for a cardinal $$\kappa>2^{\aleph_0}$$, the following two statements are equivalent:

1. There is a family $$\mathcal{A}\subseteq [\kappa]^{\aleph_1}$$ of cardinality $$>\kappa$$ with pairwise finite intersection.

2. There is a sequence $$\langle A_\alpha:\alpha<\omega_1\rangle$$ such that each $$A_\alpha$$ is a finite collection of regular cardinals from the interval $$(2^{\aleph_0},\kappa]$$, and such that for every infinite $$X\subseteq \omega_1$$, $${\rm maxpcf}(\bigcup_{\alpha\in X}A_\alpha)>\kappa$$.

The above is a theorem of Shelah (see Section 6 of [2]). It is unknown if this is consistent, but the existence of such a family entails some drastic failures of the singular cardinals hypothesis.

[1] Baumgartner, James E., Almost-disjoint sets, the dense set problem and the partition calculus, Ann. Math. Logic 9, 401-439 (1976). ZBL0339.04003.

[2] Shelah, Saharon, More on cardinal arithmetic, Arch. Math. Logic 32, No. 6, 399-428 (1993). ZBL0799.03052.

[3] Shelah, Saharon, The generalized continuum hypothesis revisited, Isr. J. Math. 116, 285-321 (2000). ZBL0955.03054.