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        Questions tagged [infinite-combinatorics]

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

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        1answer
        75 views

        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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        votes
        1answer
        97 views

        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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        1answer
        101 views

        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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        1answer
        101 views

        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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        votes
        0answers
        88 views

        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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        votes
        1answer
        111 views

        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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        1answer
        74 views

        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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        1answer
        84 views

        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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        votes
        1answer
        136 views

        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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        votes
        1answer
        260 views

        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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        votes
        1answer
        135 views

        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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        1answer
        243 views

        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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        1answer
        69 views

        $T_1$-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 ...
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        1answer
        42 views

        Maximizing set systems with property $\mathbf{B}$

        Let $X$ be an infinite set, and let ${\cal E}$ be a collection of non-empty subsets of $X$. We say that ${\cal E}$ has property $\mathbf{B}$ if there is $B\subseteq X$ such that $B\cap E\neq \emptyset$...
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        1answer
        35 views

        Connected subgraph of infinite graph with surjective homormophism, but no graph isomorphism

        For any set $X$ we set $[X]^2 = \big\{\{x,y\}: x\neq y\in X\big\}$. What is an example of a connected graph $G=(V,E)$ and a subset $S\subseteq V$ such that the subgraph $$G[S]:=(S, E\cap [S]^2)$$ is ...

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