Details

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.

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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