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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
        79 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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        75 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
        91 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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        67 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
        62 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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        1answer
        133 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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        1answer
        237 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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        1answer
        111 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
        200 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
        38 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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        1answer
        72 views

        Linear intersection number and chromatic number for infinite graphs

        Given a hypergraph $H=(V,E)$ we let its intersection graph $I(H)$ be defined by $V(I(H)) = E$ and $E(I(H)) = \{\{e,e'\}: (e\neq e'\in E) \land (e\cap e'\neq \emptyset)\}$. A linear hypergraph is a ...
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        35 views

        Minimizing the set of “faulty” edges in a map between the vertex sets of $2$ graphs

        The starting point of this question is the fact that for some simple, undirected graphs $G, H$ there is no graph homomorphism $f:G\to H$. This is the case for instance if $\chi(G)>\chi(H)$. ...
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        1answer
        42 views

        Tightly knit graphs on $\omega$

        We say that a simple, undirected graph $G = (\omega, E)$ on the vertex set $\omega$ is tightly knit if there is a positive integer $n>2$ such that for all $v,w\in \omega$ there is a cycle $C$ of ...

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