Quasiselective and weakly Ramsey ultrafilters. (English) Zbl 1518.03011

Forti, Marco

Riv. Mat. Univ. Parma (N.S.) 13, No. 1, 73-86 (2022).

There are several generalizations of selective (Ramsey) ultrafilters (on the set \(\mathbb{N}\) of natural numbers), such as quasi-selective and weakly Ramsey ultrafilters. This paper considers further variations on these notions. Recall that a function \(f:\mathbb{N}\rightarrow\mathbb{N}\) has some property modulo an ultrafilter \(\mathcal U\) (short: mod \(\mathcal U\)) if there is a set in \(\mathcal U\) on which the function has this property.
A nonprincipal ultrafilter \(\mathcal U\) on \(\mathbb{N}\) is:

–

\(f\)-quasi-selective for some \(f:\mathbb{N}\rightarrow\mathbb{N}\) (short: \(f\)-QS) if for all \(g:\mathbb{N}\rightarrow\mathbb{N}\), \(g\leq f\) mod \(\mathcal U\) implies that \(g\) is nondecreasing mod \(\mathcal U\);

–

quasi-selective if it is \(id\)-QS, where \(id\) is the identity function on \(\mathbb{N}\);

–

properly quasi-selective (short: PQS) if it is \(f\)-QS for some, but not all \(f:\mathbb{N}\rightarrow\mathbb{N}\);

–

strongly quasi-selective (short: SQS) if it is \(f\)-QS for some 1-1 function \(f\);

–

weakly quasi-selective if it is PQS, but not SQS;

–

weakly Ramsey (short: WR) if for every finite colouring \(c:[\mathbb{N}]^2\rightarrow m\) there is \(U\in{\mathcal U}\) such that \(|c[[U]^2]|\leq 2\);

–

properly weakly Ramsey (short: PWR) if it is WR but not Ramsey.

Proposition 1.2. If \(\mathcal U\) is PQS, then every function is either constant or finite-to-one mod \(\mathcal U\). Hence all PQS ultrafilters are nonselective P-points.
Proposition 1.3. Let \(\mathcal U\) be \(f\)-QS for some \(f\). Then \(\mathcal U\) is rapid if and only if it is selective.
Proposition 1.6. Assume that \(\mathcal U\) is not a Q-point and let \(f\) be nondecreasing and unbounded. Then there exists an increasing function \(\varphi\) such that \(\varphi{\mathcal U}\cong{\mathcal U}\) is not \(f\)-QS.
For a PWR ultrafilter there is an interval partition \({\mathcal P}=\{[p_n,p_{n+1}):n\in\mathbb{N}\}\) of \(\mathbb{N}\) witnessing that it is not selective (that is, there is no \(U\in{\mathcal U}\) so that \(|U\cap[p_n,p_{n+1})|\leq 1\) for all \(n\in\mathbb{N}\)). Lemma 2.1 proves that then there is \(U\in{\mathcal U}\) such that exactly one of the following cases occurs:

(i)

\(f\) is constant on \(U\);

(ii)

\(f\) is increasing on \(U\);

(iii)

\(f\) is constant on each \(U\cap[p_n,p_{n+1})\) and jumps upwards between intervals or

(iv)

\(f\) is decreasing on each \(U\cap[p_n,p_{n+1})\) and jumps upwards between intervals.

Finally, Theorems 1.5 and 2.2 and Corollary 2.3 analyze several Dedekind cuts of ultrapowers modulo \(\mathcal U\) in the sense of Blass, obtaining several equivalent conditions for the ultrafilter properties defined above.

Reviewer: Boris Šobot (Novi Sad)

MSC:

03E05	Other combinatorial set theory
03E02	Partition relations
03E65	Other set-theoretic hypotheses and axioms

Keywords:

selective ultrafilters; quasi-selective ultrafilters; weakly Ramsey ultrafilters; interval P-points

Citations:

Zbl 0305.02065; Zbl 1270.03105