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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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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 $K \in (H,G]$, $... 0answers 160 views ### ladder system uniformization at successors of singulars Shelah proved (paper 667) that if GCH holds and$\lambda$is singular, then for every stationary$S \subseteq \{ \alpha < \lambda^+ : \text{cf}(\alpha) = \text{cf}(\lambda) \}$, there is a ladder ... 0answers 94 views ### “Uniformly continuous” environment sum of a bijection$\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 $$\text{es}_f(x,y) = \sum\{f(x', y'): |(... 0answers 175 views ### Nowhere Baire spaces Studying the article "Barely Baire spaces" of W. Fleissner and K. Kunen, using stationary sets, they show an example of a Baire space whose square is nowhere Baire (we call a space X nowhere Baire ... 1answer 115 views ### Chromatic number of the linear graph on [\omega]^\omega Let [\omega]^\omega denote the set of infinite subsets of \omega. Let$$E = \{\{a,b\}: a,b\in [\omega]^\omega\text{ and } |a\cap b| = 1\}.$$It is clear that G = ([\omega]^\omega, E) has no ... 0answers 37 views ### Minimizing the set of multiply covered elements in a linear hypergraph We say that a hypergraph H=(V,E) is a linear hypergraph if it has the following properties: if e_1\neq e_2\in E then |e_1\cap e_2|\leq 1, and \bigcup E = V. We say that C\subseteq E is a ... 1answer 122 views ### Injective choice function for “lines” in an infinite cardinal Let \lambda be an infinite cardinal and suppose {\cal L} is a collection of subsets of \lambda such that |k| = \lambda for all k\in {\cal L} and, if k_1\neq k_2\in {\cal L} then |k_1\cap ... 1answer 75 views ### Dense subfilter of selective ultrafilter Given selective ultrafilter \mathcal{U} on \omega and dense filter \mathcal{F_1}=\{A\subset\omega~|~\rho(A)=1\}, where \rho(A)=\lim_{n\to\infty}\frac{|A\cap n|}{n} if the limit exists. Let \... 1answer 97 views ### Dense filter and selective ultrafilter We say that \rho(A)=\lim_{n\to\infty}\frac{|A\cap n|}{n} is the density of subset A\subset\omega if the limit exists. Let us define the filter \mathcal{F_1}=\{A\subset\omega~|~\rho(A)=1\}. ... 1answer 101 views ### Some kind of idempotence of dense filter In discussion of following questions question1, question2, question3 became clear (see definitions in question3 ) that for the Frechet filter \mathcal{N} we have \mathcal{N}\nsim\mathcal{N}\otimes\... 1answer 102 views ### Maximal elements in the Rudin-Keisler ordering Let \text{NPU}(\omega) be the set of non-principal [ultafilters][1] on \omega. The Rudin-Keisler preorder on \text{NPU}(\omega) is defined by$${\cal U} \leq_{RK} {\cal V} :\Leftrightarrow (\... 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 ... 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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