# Maximal intersecting families on $\omega$ that are not ultrafilters

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

## 1 Answer

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 $$U,V,W$$.