# Questions tagged [infinite-combinatorics]

Combinatorial properties of infinite sets. This is a corner-point of set theory and combinatorics.

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### Existance of bijective function which maps tensor product of subsets of a selective ultrafilter into the ultrafilter

In the answer on this question Andreas Blass had shown that for any selective ultrafilter $\scr{U}$ on $\omega$ and for any free subfilter $\scr{F}\subset{U}$ doesn't exist bijection $\varphi:\omega^2\...

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### Dense subfilter of selective ultrafilter

Given selective ultrafilter $\mathcal{U}$ on $\omega$ and dense filter $\mathcal{F_1}=\{A\subset\omega~|~\rho(A)=1\}$, where $\rho(A)=\lim_{n\to\infty}\frac{|A\cap n|}{n}$ if the limit exists. Let $\...

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### Dense filter and selective ultrafilter

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 $\mathcal{F_1}=\{A\subset\omega~|~\rho(A)=1\}$.
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### Largest subset of the powerset of a countable set in which no set includes another [duplicate]

Let S be a set that has countably-infinitely many members. Let a subset of $\mathcal{P}(S)$ (the power-set of S) have the Sperner-family-property iff no two of its members are such that one of them is ...

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### Some kind of idempotence of dense filter

In discussion of following questions question1, question2, question3 became clear (see definitions in question3 ) that for the Frechet filter $\mathcal{N}$ we have $\mathcal{N}\nsim\mathcal{N}\otimes\...

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### Maximal elements in the Rudin-Keisler ordering

Let $\text{NPU}(\omega)$ be the set of non-principal [ultafilters][1] on $\omega$. The Rudin-Keisler preorder on $\text{NPU}(\omega)$ is defined by
$${\cal U} \leq_{RK} {\cal V} :\Leftrightarrow (\...

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### Covering numbers - looking for a more combinatorial proof

For cardinals $\mu$, $\kappa$, $\theta$, and $\sigma$, the covering number $cov(\mu,\kappa,\theta,\sigma)$ is defined to be the minimum cardinality of a set $P\subseteq [\mu]^{<\kappa}$ such that ...

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### Minimal cardinality of a filter base of a non-principal uniform ultrafilters

Let $\kappa$ be an infinite cardinal. An ultrafilter ${\cal U}$ on $\kappa$ is said to be uniform if $|R|=\kappa$ for all $R\in{\cal U}$. If ${\cal U}$ is a non-principal ultrafilter on $\kappa$, ...

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### Maximizing “happy” vertices in splitting an infinite graph

This question is motivated by a real life task (which is briefly described after the question.)
Let $G=(V,E)$ be an infinite graph. For $v\in V$ let $N(v) = \{x\in V: \{v,x\}\in E\}$. If $S\subseteq ...

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### The example of the idempotent filter or subsets family with finite intersections property

From the answer of Andreas Blass and comments of Ali Enayat on my question Selective ultrafilter and bijective mapping it became clear that for any free ultrafilter $\mathcal{U}$ we have $\mathcal{U}\...

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### Graphs with minimum degree $\delta(G)\lt\aleph_0$

Let $G=(V,E)$ be a graph with minimum degree $\delta(G)=n\lt\aleph_0$. Does $G$ necessarily have a spanning subgraph $G'=(V,E')$ which also has minimum degree $\delta(G')=n$ and is minimal with that ...

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### On infinite combinatorics of ultrafilters

Let $\mathcal{U}$ be (selective) ultrafilter and $\omega = \coprod_{i<\omega}S_i$ be partition of $\omega$ with small sets $S_i\notin\mathcal{U}$. All $S_i$ are infinite. Does there exist a system ...

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### Are two “perfectly dense” hypergraphs on $\mathbb{N}$ necessarily isomorphic?

We say that a hypergraph $(\mathbb{N}, E)$ where $E\subseteq {\cal P}(\mathbb N)$ is perfectly dense if
$\mathbb{N}\notin E$,
all $e\in E$ are infinite,
$e_1, e_2 \in E$ implies $|e_1\cap e_2| = 1$,...

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### Selective ultrafilter and bijective mapping

For arbitrary (selective) ultrafilter $\mathcal{F}$ does there exist bijection $\phi:[\omega]^2\to\omega$ with the property: $\forall B\in\mathcal{F} : \phi([B]^2)\in\mathcal{F}$ ?

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### $T_1$-spaces vs $T_1$-hypergraphs

Let us say that a hypergraph $H=(V,E)$ is $T_1$ if for $x\neq y$ there is $e\in E$ such that $e\cap\{x,y\} = \{x\}$.
Note that for any $T_1$-space $(X,\tau)$ the topology $\tau$ contains the ...