410 research outputs found

    On the Duality of Semiantichains and Unichain Coverings

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    We study a min-max relation conjectured by Saks and West: For any two posets PP and QQ the size of a maximum semiantichain and the size of a minimum unichain covering in the product P×QP\times Q are equal. For positive we state conditions on PP and QQ that imply the min-max relation. Based on these conditions we identify some new families of posets where the conjecture holds and get easy proofs for several instances where the conjecture had been verified before. However, we also have examples showing that in general the min-max relation is false, i.e., we disprove the Saks-West conjecture.Comment: 10 pages, 3 figure

    On Saturated kk-Sperner Systems

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    Given a set XX, a collection FP(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 FP(X)\mathcal{F}\subseteq\mathcal{P}(X) is 2k12^{k-1}. We disprove this conjecture by showing that there exists ε>0\varepsilon>0 such that for every kk and Xn0(k)|X| \geq n_0(k) there exists a saturated kk-Sperner system FP(X)\mathcal{F}\subseteq\mathcal{P}(X) with cardinality at most 2(1ε)k2^{(1-\varepsilon)k}. A collection FP(X)\mathcal{F}\subseteq \mathcal{P}(X) is said to be an oversaturated kk-Sperner system if, for every SP(X)FS\in\mathcal{P}(X)\setminus\mathcal{F}, F{S}\mathcal{F}\cup\{S\} contains more chains of length k+1k+1 than F\mathcal{F}. Gerbner et al. proved that, if Xk|X|\geq k, then the smallest such collection contains between 2k/212^{k/2-1} and O(logkk2k)O\left(\frac{\log{k}}{k}2^k\right) elements. We show that if Xk2+k|X|\geq k^2+k, then the lower bound is best possible, up to a polynomial factor.Comment: 17 page

    The leafage of a chordal graph

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    The leafage l(G) of a chordal graph G is the minimum number of leaves of a tree in which G has an intersection representation by subtrees. We obtain upper and lower bounds on l(G) and compute it on special classes. The maximum of l(G) on n-vertex graphs is n - lg n - (1/2) lg lg n + O(1). The proper leafage l*(G) is the minimum number of leaves when no subtree may contain another; we obtain upper and lower bounds on l*(G). Leafage equals proper leafage on claw-free chordal graphs. We use asteroidal sets and structural properties of chordal graphs.Comment: 19 pages, 3 figure

    Chains, Antichains, and Complements in Infinite Partition Lattices

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    We consider the partition lattice Πκ\Pi_\kappa on any set of transfinite cardinality κ\kappa and properties of Πκ\Pi_\kappa whose analogues do not hold for finite cardinalities. Assuming the Axiom of Choice we prove: (I) the cardinality of any maximal well-ordered chain is always exactly κ\kappa; (II) there are maximal chains in Πκ\Pi_\kappa of cardinality >κ> \kappa; (III) if, for every cardinal λ<κ\lambda < \kappa, we have 2λ<2κ2^{\lambda} < 2^\kappa, there exists a maximal chain of cardinality <2κ< 2^{\kappa} (but κ\ge \kappa) in Π2κ\Pi_{2^\kappa}; (IV) every non-trivial maximal antichain in Πκ\Pi_\kappa has cardinality between κ\kappa and 2κ2^{\kappa}, and these bounds are realized. Moreover we can construct maximal antichains of cardinality max(κ,2λ)\max(\kappa, 2^{\lambda}) for any λκ\lambda \le \kappa; (V) all cardinals of the form κλ\kappa^\lambda with 0λκ0 \le \lambda \le \kappa occur as the number of complements to some partition PΠκ\mathcal{P} \in \Pi_\kappa, and only these cardinalities appear. Moreover, we give a direct formula for the number of complements to a given partition; (VI) Under the Generalized Continuum Hypothesis, the cardinalities of maximal chains, maximal antichains, and numbers of complements are fully determined, and we provide a complete characterization.Comment: 24 pages, 2 figures. Submitted to Algebra Universalis on 27/11/201

    Set Systems Containing Many Maximal Chains

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    The purpose of this short problem paper is to raise an extremal question on set systems which seems to be natural and appealing. Our question is: which set systems of a given size maximise the number of (n+1)(n+1)-element chains in the power set P({1,2,,n})\mathcal{P}(\{1,2,\dots,n\})? We will show that for each fixed α>0\alpha>0 there is a family of α2n\alpha 2^n sets containing (α+o(1))n!(\alpha+o(1))n! such chains, and that this is asymptotically best possible. For smaller set systems we are unable to answer the question. We conjecture that a `tower of cubes' construction is extremal. We finish by mentioning briefly a connection to an extremal problem on posets and a variant of our question for the grid graph.Comment: 5 page

    A scattering of orders

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    A linear ordering is scattered if it does not contain a copy of the rationals. Hausdorff characterised the class of scattered linear orderings as the least family of linear orderings that includes the class B \mathcal B of well-orderings and reversed well-orderings, and is closed under lexicographic sums with index set in B \mathcal B. More generally, we say that a partial ordering is κ \kappa -scattered if it does not contain a copy of any κ \kappa -dense linear ordering. We prove analogues of Hausdorff's result for κ \kappa -scattered linear orderings, and for κ \kappa -scattered partial orderings satisfying the finite antichain condition. We also study the Qκ \mathbb{Q}_\kappa -scattered partial orderings, where Qκ \mathbb{Q}_\kappa is the saturated linear ordering of cardinality κ \kappa , and a partial ordering is Qκ \mathbb{Q}_\kappa -scattered when it embeds no copy of Qκ \mathbb{Q}_\kappa . We classify the Qκ \mathbb{Q}_\kappa -scattered partial orderings with the finite antichain condition relative to the Qκ \mathbb{Q}_\kappa -scattered linear orderings. We show that in general the property of being a Qκ \mathbb{Q}_\kappa -scattered linear ordering is not absolute, and argue that this makes a classification theorem for such orderings hard to achieve without extra set-theoretic assumptions

    On Ordinal Invariants in Well Quasi Orders and Finite Antichain Orders

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    We investigate the ordinal invariants height, length, and width of well quasi orders (WQO), with particular emphasis on width, an invariant of interest for the larger class of orders with finite antichain condition (FAC). We show that the width in the class of FAC orders is completely determined by the width in the class of WQOs, in the sense that if we know how to calculate the width of any WQO then we have a procedure to calculate the width of any given FAC order. We show how the width of WQO orders obtained via some classical constructions can sometimes be computed in a compositional way. In particular, this allows proving that every ordinal can be obtained as the width of some WQO poset. One of the difficult questions is to give a complete formula for the width of Cartesian products of WQOs. Even the width of the product of two ordinals is only known through a complex recursive formula. Although we have not given a complete answer to this question we have advanced the state of knowledge by considering some more complex special cases and in particular by calculating the width of certain products containing three factors. In the course of writing the paper we have discovered that some of the relevant literature was written on cross-purposes and some of the notions re-discovered several times. Therefore we also use the occasion to give a unified presentation of the known results
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