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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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      4
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      2answers
      393 views

      “Rocket elements” in bijections $f:\mathbb{N}\to \mathbb{N}$

      Let $\mathbb{N}$ denote the set of non-negative integers. If $f:\mathbb{N}\to\mathbb{N}$ is a map, we set for $k\in \mathbb{N}$: $f^{(0)}(k) = k$, and $f^{(n+1)}(k) = f(f^{(n)}(k))$ for all $n\in\...
      3
      votes
      1answer
      63 views

      Induced subgraphs of $\text{Exp}(G, K_2)$

      If $G, H$ are simple, undirected graphs, we define the exponential graph $\text{Exp}(G,H)$ to be the following graph: the vertex set is the set of all maps $f:V(G)\to V(H)$ two maps $f\neq g: V(G)\to ...
      3
      votes
      0answers
      131 views

      Infinite group generated by a single coset

      Let $G$ be an infinite countable group having a core-free subgroup $H$ such that the interval $[H,G]$ in the subgroup lattice $\mathcal{L}(G)$ is ACC of infinite length, and for every $K \in (H,G]$, $...
      7
      votes
      0answers
      160 views

      ladder system uniformization at successors of singulars

      Shelah proved (paper 667) that if GCH holds and $\lambda$ is singular, then for every stationary $S \subseteq \{ \alpha < \lambda^+ : \text{cf}(\alpha) = \text{cf}(\lambda) \}$, there is a ladder ...
      4
      votes
      0answers
      94 views

      “Uniformly continuous” environment sum of a bijection $\varphi:\mathbb{Z}\times \mathbb{Z} \to \mathbb{Z}$

      Given any function $f: \mathbb{Z}\times \mathbb{Z}\to \mathbb{Z}$ we define the environment sum of $(x,y)\in\mathbb{Z}\times \mathbb{Z}$ with respect to $f$ by $$\text{es}_f(x,y) = \sum\{f(x', y'): |(...
      3
      votes
      0answers
      175 views

      Nowhere Baire spaces

      Studying the article "Barely Baire spaces" of W. Fleissner and K. Kunen, using stationary sets, they show an example of a Baire space whose square is nowhere Baire (we call a space $X$ nowhere Baire ...
      2
      votes
      1answer
      115 views

      Chromatic number of the linear graph on $[\omega]^\omega$

      Let $[\omega]^\omega$ denote the set of infinite subsets of $\omega$. Let $$E = \{\{a,b\}: a,b\in [\omega]^\omega\text{ and } |a\cap b| = 1\}.$$ It is clear that $G = ([\omega]^\omega, E)$ has no ...
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      0answers
      37 views

      Minimizing the set of multiply covered elements in a linear hypergraph

      We say that a hypergraph $H=(V,E)$ is a linear hypergraph if it has the following properties: if $e_1\neq e_2\in E$ then $|e_1\cap e_2|\leq 1$, and $\bigcup E = V$. We say that $C\subseteq E$ is a ...
      2
      votes
      1answer
      122 views

      Injective choice function for “lines” in an infinite cardinal

      Let $\lambda$ be an infinite cardinal and suppose ${\cal L}$ is a collection of subsets of $\lambda$ such that $|k| = \lambda$ for all $k\in {\cal L}$ and, if $k_1\neq k_2\in {\cal L}$ then $|k_1\cap ...
      2
      votes
      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 $\...
      3
      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 $\mathcal{F_1}=\{A\subset\omega~|~\rho(A)=1\}$. ...
      1
      vote
      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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      votes
      1answer
      102 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 (\...
      2
      votes
      0answers
      90 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 ...
      3
      votes
      1answer
      125 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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