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A family ${\cal S}\subseteq{\cal P}(\omega)$ is intersecting if any two members of ${\cal S}$ have non-empty intersection. Zorn's Lemma implies that every intersecting family is contained in a maximal intersecting family with respect to set inclusion $\subseteq$.

Every ultrafilter on $\omega$ is maximal intersecting. Let ${\frak U}$ be the collection of ultrafilters on $\omega$, and let ${\frak M}$ be the collection of maximal intersecting families on $\omega$. We have ${\frak U}\subseteq {\frak M}$ and $|{\frak U}| = 2^{2^{\aleph_0}}$.

Question. Is $|{\frak M}\setminus{\frak U}| = 2^{2^{\aleph_0}}$?

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1 Answer 1

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Let $U,V,W$ be three distinct ultrafilters on $\omega$. Let $M$ be the family of those subsets of $\omega$ that belong to at least two of $U,V,W$. Then $M$ is a maximal intersecting family, it is not an ultrafilter, and we can get $2^{2^{\aleph_0}}$ such $M$'s by choosing different $U,V,W$.

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