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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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        3
        votes
        2answers
        101 views

        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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        1answer
        35 views

        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 $N_D = V$. ...
        3
        votes
        0answers
        223 views

        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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        0answers
        31 views

        Giant component in continuum random graph

        Is there some similar statement to Giant component theorem for the infinite graph (particularly with continuum vertices)?
        5
        votes
        0answers
        125 views

        Consistency of monochromatic uniformization at an inaccessible cardinal

        Let $\kappa$ be an inaccessible cardinal, is the following uniformization principle at $\kappa$ consistent (is it consistent with GCH?): there exists a ladder system $\langle A_\alpha\subset \alpha: \...
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        votes
        2answers
        463 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
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        1answer
        68 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
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        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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        0answers
        174 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 ...
        5
        votes
        1answer
        174 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
        178 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
        117 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
        39 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
        123 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 $\...

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