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        All Questions

        2
        votes
        1answer
        109 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 ...
        5
        votes
        3answers
        156 views

        Disjunction number of a graph

        Let $S\neq \emptyset$ be a set. We make its powerset ${\cal P}(S)$ into a simple, undirected graph by saying that $A, B\in{\cal P}(S)$ form an edge if and only if $A\cap B=\emptyset$. The ...
        1
        vote
        1answer
        76 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 ...
        1
        vote
        0answers
        67 views

        Generalization of the linear extension theorem to directed acyclic graphs

        Using Zorn's lemma one can prove a generalization of the order extension theorem, that states any acyclic digraph is always contained in another acyclic unilaterally connected digraph on the same ...
        0
        votes
        1answer
        73 views

        Linear intersection number and chromatic number for infinite graphs

        Given a hypergraph $H=(V,E)$ we let its intersection graph $I(H)$ be defined by $V(I(H)) = E$ and $E(I(H)) = \{\{e,e'\}: (e\neq e'\in E) \land (e\cap e'\neq \emptyset)\}$. A linear hypergraph is a ...
        3
        votes
        1answer
        206 views

        Does every directed graph have a directed coloring with $4$ colors?

        Every finite directed graph has a majority coloring with $4$ colors. (The notion of majority coloring is defined below.) Question. Can every infinite directed graph be majority-colored with $4$ ...
        2
        votes
        1answer
        138 views

        Induced minors of $\{0,1\}^\omega$

        Let $G=(V,E)$ be a simple, undirected graph. Suppose that ${\cal S}$ is a collection of non-empty, connected, and pairwise disjoint subsets of $V$. Let $G({\cal S})$ be the graph with vertex set ${\...
        -1
        votes
        1answer
        75 views

        Finding a good transversal basis

        A hypergraph $H=(V,E)$ consists of an non-empty set $V$ and a collection $E\subseteq {\cal P}(V)\setminus \{\emptyset\}$ of non-empty subsets of $V$. A transversal of $H$ is a set $T\subseteq V$ such ...
        0
        votes
        1answer
        85 views

        Connected infinite graphs in which all matchings are “small”

        Is there a countable, simple, connected graph $G=(\omega, E)$ such that $\text{deg}(v)$ is infinite for all $v\in \omega$, and for all matchings $M\subseteq E$ the set $V\setminus (\bigcup M)$ is ...
        12
        votes
        1answer
        387 views

        Is each cover of the plane by lines minimizable?

        A cover $\mathcal C$ of a set $X$ by subsets of $X$ is called $\bullet$ minimal if for every $C\in\mathcal C$ the family $\mathcal C\setminus\{C\}$ is not a cover of $X$; $\bullet$ minimizable if $\...
        34
        votes
        2answers
        3k views

        How to find Erd?s' treasure trove?

        The renowned mathematician, Paul Erd?s, has published more than 1500 papers in various branches of mathematics including discrete mathematics, graph theory, number theory, mathematical analysis, ...
        5
        votes
        1answer
        161 views

        “König's theorem” for $T_2$-spaces?

        For any topological space $(X,\tau)$ we define a matching to be a collection of non-empty and pairwise disjoint open sets. We define the matching number $\nu(X,\tau)$ to be the smallest cardinal $\...
        1
        vote
        2answers
        172 views

        Bipartite subgraphs with lots of edges

        Suppose $G=(V,E)$ is a simple, undirected graph with $|V|,|E|$ infinite. Is there $B\subseteq E$ with $|B| = |E|$ such that $(V,B)$ is bipartite?
        3
        votes
        1answer
        170 views

        Infinite graph with lots of non-isomorphic induced subgraphs

        Given an infinite cardinal $\kappa$, is there a graph on $\kappa$ vertices that contains $2^\kappa$ pairwise non-isomorphic induced subgraphs?
        8
        votes
        1answer
        395 views

        Does Vizing's conjecture hold for the infinite graphs?

        In finite graph theory, there are many (in)equalities which relate the integer value of a certain graph invariant (e.g. domination or chromatic number) for the product of two finite graphs (e.g. ...

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