We say that $\rho(A)=\lim_{n\to\infty}\frac{|A\cap n|}{n}$ is the density of subset $A\subset\omega$ if the limit exists. Let us define the filter $\mathcal{F_1}=\{A\subset\omega~|~\rho(A)=1\}$.

Question: Does there exist (in ZFC & CH) selective ultrafilter $\mathcal{U}$ and a bijection $\varphi:\omega\to\omega$ such that $\varphi(\mathcal{F_1})\subset\mathcal{U}$ ?

Remark: Someone had downvoted two similar questions. Please, explain what is wrong if something is wrong

  • $\begingroup$ I would assume that people downvoted because they missed the word "selective". This turns the question from a triviality into something pretty difficult. Please, read carefully before you downvote! $\endgroup$ – Jan-Christoph Schlage-Puchta Mar 26 '19 at 20:37

No, there are no such $\mathcal U$ and $\varphi$.

Let me start the proof with two simplifying observations. First, $\varphi$ is irrelevant, because bijections of $\omega$ to itself preserve selectivity of ultrafilters.

Second, every selective ultrafilter $\mathcal U$ is a P-point, which means that, given any countably many sets $A_n\in\mathcal U$, there is a set $B\in\mathcal U$ almost included in all of the $A_n$"s, i.e., $B-A_n$ is finite for all $n$. To prove that selective ultrafilters $\mathcal U$ have this property, let sets $A_n\in\mathcal U$ be given. If $\bigcap_nA_n\in\mathcal U$, then this intersection serves as the required $B$, so assume $\bigcap_nA_n\notin\mathcal U$. Then we can partition $\omega$ into the pieces $\omega-A_0, A_0-A_1, A_1-A_2, A_2-A_3, \dots$ and $\bigcap_nA_n$, none of which are in $\mathcal U$. Selectivity provides a set $B\in\mathcal U$ that intersects each of these pieces in at most one point. Then $B-A_n$ is finite (in fact it has at most $n+1$ elements) for each $n$, as required.

So now it suffices to show that no P-point ultrafilter $\mathcal U$ can include the density-$1$ filter $\mathcal F_I$. For each natural number $n\geq 2$, partition $\omega$ into $n$ sets each of which has density $\frac1n$; for example, take the congruence classes modulo $n$. Being an ultrafilter, $\mathcal U$ must contain one of these $n$ sets, say $A_n$. If $\mathcal U$ is a P-point, it contains a set $B$ almost included in all of these $A_n$'s. But from almost inclusion it easily follows that $B$ has upper density at most the density $\frac1n$ of $A_n$. (Upper density is defined like density but with $\limsup$ instead of $\lim$.) So $B$ has upper density $0$, which means it has density $0$. Thus, $\omega-B$ has density $1$ and is therefore in $\mathcal F_I$, but it is not in $\mathcal U$ because $B\in\mathcal U$.

  • 1
    $\begingroup$ Thank you. Really, $\varphi(\mathcal{F_1})\subset\mathcal{U}\implies\mathcal{F_1}\subset\varphi^{-1}(\mathcal{U})$ which is impossible because $\varphi^{-1}(\mathcal{U})$ is also selective an thus contains sets with density 0. Weak question. But what about this one? $\endgroup$ – ar.grig Mar 24 '19 at 3:11

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