Riv. Mat. Univ. Parma, Vol. 13, No. 1, 2022
Marco Forti [a]
Quasiselective and weakly Ramsey ultrafilters
Pages: 73-86
Received: 6 May 2021
Accepted: 22 July 2021
Mathematics Subject Classification: 03E02, 03E05, 03E20, 03E65.
Keywords: Selective ultrafilters, quasi-selective ultrafilters, weakly Ramsey ultrafilters, interval P-points.
Authors address:
[a]: University of Pisa, Depart. Mathematics, Largo B. Pontecorvo 5, 56127 Pisa, Italy.
Dedicated to Roberto Dvornicich on his seventieth birthday
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Abstract:
Selective (Ramsey) ultrafilters are characterized by many equivalent
properties.
Natural weakenings of these properties led to the inequivalent notions
of weakly Ramsey and of quasi-selective ultrafilter, introduced and
studied in [1] and [4], respectively. Call
\(\,\mathcal U\) weakly Ramsey
if for
every finite colouring of \([ \mathbb N ]^{2}\) there is \(\,U\in\mathcal U\,\)
s.t. \([U]^{2}\)
has only two colours, and call
\(\,\mathcal U\) \(f\)-quasi-selective if every function
\(g\le f\) is nondecreasing on some \(U\in\mathcal U\).
(So the quasi-selective ultrafilters of [4] are
\(id\)-quasi selective.)
In this paper we characterize
those weakly Ramsey ultrafilters that are isomorphic to a
quasi-selective ultrafilter by analyzing the relations between various natural cuts
of the ultrapowers of \( \mathbb N \) modulo these ultrafilters.
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