544 research outputs found

    Exact Bounds for Some Hypergraph Saturation Problems

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    Let W_n(p,q) denote the minimum number of edges in an n x n bipartite graph G on vertex sets X,Y that satisfies the following condition; one can add the edges between X and Y that do not belong to G one after the other so that whenever a new edge is added, a new copy of K_{p,q} is created. The problem of bounding W_n(p,q), and its natural hypergraph generalization, was introduced by Balogh, Bollob\'as, Morris and Riordan. Their main result, specialized to graphs, used algebraic methods to determine W_n(1,q). Our main results in this paper give exact bounds for W_n(p,q), its hypergraph analogue, as well as for a new variant of Bollob\'as's Two Families theorem. In particular, we completely determine W_n(p,q), showing that if 1 <= p <= q <= n then W_n(p,q) = n^2 - (n-p+1)^2 + (q-p)^2. Our proof applies a reduction to a multi-partite version of the Two Families theorem obtained by Alon. While the reduction is combinatorial, the main idea behind it is algebraic

    On Saturated kk-Sperner Systems

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    Given a set XX, a collection FβŠ†P(X)\mathcal{F}\subseteq\mathcal{P}(X) is said to be kk-Sperner if it does not contain a chain of length k+1k+1 under set inclusion and it is saturated if it is maximal with respect to this property. Gerbner et al. conjectured that, if ∣X∣|X| is sufficiently large with respect to kk, then the minimum size of a saturated kk-Sperner system FβŠ†P(X)\mathcal{F}\subseteq\mathcal{P}(X) is 2kβˆ’12^{k-1}. We disprove this conjecture by showing that there exists Ξ΅>0\varepsilon>0 such that for every kk and ∣X∣β‰₯n0(k)|X| \geq n_0(k) there exists a saturated kk-Sperner system FβŠ†P(X)\mathcal{F}\subseteq\mathcal{P}(X) with cardinality at most 2(1βˆ’Ξ΅)k2^{(1-\varepsilon)k}. A collection FβŠ†P(X)\mathcal{F}\subseteq \mathcal{P}(X) is said to be an oversaturated kk-Sperner system if, for every S∈P(X)βˆ–FS\in\mathcal{P}(X)\setminus\mathcal{F}, Fβˆͺ{S}\mathcal{F}\cup\{S\} contains more chains of length k+1k+1 than F\mathcal{F}. Gerbner et al. proved that, if ∣X∣β‰₯k|X|\geq k, then the smallest such collection contains between 2k/2βˆ’12^{k/2-1} and O(log⁑kk2k)O\left(\frac{\log{k}}{k}2^k\right) elements. We show that if ∣X∣β‰₯k2+k|X|\geq k^2+k, then the lower bound is best possible, up to a polynomial factor.Comment: 17 page

    Not All Saturated 3-Forests Are Tight

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    A basic statement in graph theory is that every inclusion-maximal forest is connected, i.e. a tree. Using a definiton for higher dimensional forests by Graham and Lovasz and the connectivity-related notion of tightness for hypergraphs introduced by Arocha, Bracho and Neumann-Lara in, we provide an example of a saturated, i.e. inclusion-maximal 3-forest that is not tight. This resolves an open problem posed by Strausz
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