(Joint question with Piotr Szewczak.)

**Definitions and notation.** By *filter* we mean a filter on $\omega$ containing the cofinite sets at least.

For a filter $\mathcal{F}$, let $\mathcal{F}^+:=\{A\subseteq\omega : A^c\notin \mathcal{F}\}$.

For an infinite set $A\subseteq\omega$ and a natural number $n$, define $\mathrm{next}(A,n)=\min\{k\in A:n<k\}$.

For natural numbers $m<n$, let $[m,n]:=\{m,m+1,\dots,n\}$.

**The question.** Which filters $\mathcal{F}$ we have the following property?

- For each $A\in \mathcal{F}^+$ and all $B\in \mathcal{F}$, the set $C := \bigcup\{[n,\mathrm{next}(A,n)] : n \in A\cap B\}$ is in $\mathcal{F}$.

The Frechet filter (of all cofinite sets), and every ultrafilter, have this proeprty.

In particular, are there concrete examples of filters that are not Frechet and not uf (and better not the above under a finite-to-one map) but have this property?

**Update.** Blass and Brian provide below examples that are ultrafilters under finite-to-one maps. Our research developed in the meanwhile to see that we need examples that cannot be mapped to the Frechet filter or an ultrafilter by a finite-to-one function. Consistently, as Blass points out, there are no such examples (Filter Dichotomy), so the question is, for example, *what can be said if CH holds*? More concretely, how prevalent are such examples under CH? Are there "important" examples (i.e., not ones cooked up by transfinite induction for the purpose of the question).

Any other axioms relevant for a positive answer?

lackingyour property, but, just in case, here's one: For $i=0,1,2$, let $A_i=\{n\in\omega:n\equiv i\pmod3\}$. Let $\mathcal U$ and $\mathcal V$ be non-principal ultrafilters containing $A_0$ and $A_2$, respectively, and let $\mathcal F=\mathcal U\cap\mathcal V$, so $\mathcal F^+=\mathcal U\cup\mathcal V$. I claim this $\mathcal F$ does not have the property you described. The counterexample is $A=A_0\cup A_1$ and $B=A_0\cup A_2$. Then $A\cap B=A_0$ and $\bigcup\{[n,\text{next}(A,n)]:n\in A\cap B\}=A_0\cup A_1\notin\mathcal F$. $\endgroup$