Definition 1. A ultrafilter $\mathcal U$ on $\omega$ is called discrete (resp. nowhere dense) if for any injective map $f:\omega\to \mathbb R$ there is a set $U\in\mathcal U$ whose image $f(U)$ is a discrete (resp. nowhere dense) subspace of the real line $\mathbb R$.

It is easy to see that each p-point is a discrete ultrafilter and each discrete ultrafilter is nowhere dense. By a known result of Shelah, there are models of ZFC without nowhere dense ultrafilters.

I am interested in the ZFC-existence of asymptotic counterparts of discrete ultrafilters.

A subset $D$ of a metric space $(X,d)$ is called asymptotically discrete if for any $r\in\mathbb N$ there exists a bounded set $B\subset X$ such that $d(x,y)\ge r$ for any distinct points $x,y\in D\setminus B$.

In particular, a subset $D$ of $\mathbb Z$ is asymptotically discrete iff for every $n\in\omega$ there exists a finite set $B\subset\mathbb Z$ such that $|x-y|\ge n$ for any distinct points $x,y\in D\setminus B$.

Definition 2. A ultrafilter $\mathcal U$ on $\omega$ is called asymptotically discrete if for every injective map $f:\omega\to\mathbb Z$ there exists a set $U\in\mathcal U$ whose image $f(U)$ is asymptotically discrete in the metric space $\mathbb Z$ endowed with the standard Euclidean metric.

In this paper Petrenko and Protasov proved that an ultrafilter on $\omega$ is asymptotically discrete if it is a $p$-point or a $q$-point. They also proved that under CH there are asymptotically discrete ultrafilters which are neither $p$-points nor $q$-points.

Problem 1. Does an asymptotically discrete ultrafilter exist in ZFC?

Remark 1. If there exists a model of ZFC without asymptotically discrete ultrafilters, then in this model there is no $p$-points and no $q$-points. The existence of such a model is a well-known open problem in Set Theory. So, if Problem 1 has a simple answer, then this answer is affirmative. Maybe Kunen's OK-ultrafilters are asymptotically discrete?

Remark 2. Any ultrafilter $\mathcal U$ containing the filter of subsets of Banach density 1 in $\omega$ is not asymptotically discrete (as every subset $U\in\mathcal U$ has non-zero Banach density).


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