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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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### Avoiding multiply covered vertices in graph edge coverings

Let $G=(V,E)$ be a simple, undirected graph with $\bigcup = E$ (that is, there are no isolated vertices). We say that $C\subseteq E$ is an edge cover of $G$ if $\bigcup C = V$. For any edge cover $C$ ...
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### Dominating vertex sets in hypergraphs

Let $H=(V,E)$ be a hypergraph such that $\bigcup E = V$. For $D\subseteq V$ we set $N_D = \bigcup\{e\in E: D\cap e\neq \emptyset\}$. We say that $D\subseteq V$ is dominating if $N_D = V$. ...
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### Almost disjoint families on $\omega_1$

Suppose there is a family of $\aleph_3$ unbounded subsets of $\omega_1$ in which no set is contained in a countable union of other sets. Must there exist a (mod countable) almost disjoint family of ...
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### Giant component in continuum random graph

Is there some similar statement to Giant component theorem for the infinite graph (particularly with continuum vertices)?
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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 $f\neq g: V(G)\to ... 0answers 136 views ### Infinite group generated by a single coset Let$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]$,$...
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### 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 ...
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### 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 ...
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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}$if the limit exists. Let$\...

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