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It is well known that there are $2^{2^{\aleph_0}}$ many non-principal ultrafilters on $\omega$. … Is there a set ${\frak U}$ of non-principal ultrafilters on $\omega$ with $|{\frak U}| = 2^{2^{\aleph_0}}$ such that for ${\cal U}_1\neq {\cal U}_2\in{\frak U}$ the partially ordered sets $({\cal U}_1, …

If ${\cal U}$ is any ultrafilter on $\omega$, the pair $(\omega,{\cal U}\cup \{\emptyset\})$ is a connected topological space. Is there a non-principal ultrafilter ${\cal U}$ on $\omega$ such that we …

If ${\scr U}$ and ${\scr V}$ are ultrafilters on non-empty sets $A$ and $B$ respectively, then the tensor product ${\scr U}\otimes{\scr V}$ is the following ultrafilter on $A\times B$: $$\big\{X\subseteq …

Let ${\cal U}$ be a non-principal ultrafilter on $\omega$, and for each $n\in\omega$, let $p_n$ denote the $n$th prime, that is $p_0 = 2, p_1=3, \ldots$ Next we introduce the following standard equiva …

Let $\kappa\geq \aleph_0$ be a cardinal, and suppose that ${\cal U}$ is a non-principal ultrafilter on $\kappa$. We regard ${\cal U}$ as a poset $({\cal U}, \subseteq)$. Suppose that there are posets …

A $P$-point is an ultrafilter $\scr U$on $\omega$ such that for every function $f:\omega\to\omega$ there is $x\in {\scr U}$ such that the restriction $f|_x$ is either constant, or finite-to-one. A $Q …

(Ramsey ultrafilters do not necessarily exist.) Question. Does $\text{NPU}(\omega)/\simeq_{RK}$ have maximal elements? …

Let $\beta\omega$ be collection of all ultrafilters on $\omega$ (principal and non-principal). We endow $\beta\omega$ with an operation $+$ in the following way. …

If ${\cal U, V}$ are non-principal uniform ultrafilters on $\kappa$, do we necessarily have $b({\cal U}) = b({\cal V})$? … Thanks to Joseph van Name for making me aware of uniform ultrafilters and the fact that this question is only (potentially) interesting when restricted to these. …

Let ${\frak U}$ be the collection of ultrafilters on $\omega$, and let ${\frak M}$ be the collection of maximal intersecting families on $\omega$. …

$\DeclareMathOperator{\RK}{\mathrm{RK}}$Let $\beta\omega$ be the Stone-Cech compactification of $\omega$ with the discrete topology. We can endow $\beta\omega$ with an addition operation that extends …

Let $\NPU(\omega)$ be the set of non-principal ultrafilters on $\omega$. …

If ${\cal U}$ and ${\cal V}$ are ultrafilters on non-empty sets $A$ and $B$ respectively, then the tensor product ${\cal U}\otimes{\cal V}$ is the following ultrafilter on $A\times B$: $$\big\{X\subseteq … We say an ultrafilter ${\cal Z}$ on $\omega$ is Tensor-representable if there are non-Keisler-Rudin-equivalent ultrafilters ${\cal U}, {\cal V}$ and a bijection $\psi:\omega^2\to \omega$ such that ${\cal …

Let ${\cal U}$ be a free ultrafilter on $\omega$. Is the linearly ordered set $(\omega+1)^\omega/{\cal U}$ complete?

Let $\text{FrU}(\omega)$ be the collection of free ultrafilters on $\omega$. …