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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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      46 views

      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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      1answer
      67 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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      1answer
      92 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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      50 views

      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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      1answer
      95 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
      96 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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      0answers
      85 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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      1answer
      101 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
      72 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
      83 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
      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
      259 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
      130 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
      235 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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