551 research outputs found

    A note on blockers in posets

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    The blocker AA^{*} of an antichain AA in a finite poset PP is the set of elements minimal with the property of having with each member of AA a common predecessor. The following is done: 1. The posets PP for which A=AA^{**}=A for all antichains are characterized. 2. The blocker AA^* of a symmetric antichain in the partition lattice is characterized. 3. Connections with the question of finding minimal size blocking sets for certain set families are discussed

    Colouring set families without monochromatic k-chains

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    A coloured version of classic extremal problems dates back to Erd\H{o}s and Rothschild, who in 1974 asked which nn-vertex graph has the maximum number of 2-edge-colourings without monochromatic triangles. They conjectured that the answer is simply given by the largest triangle-free graph. Since then, this new class of coloured extremal problems has been extensively studied by various researchers. In this paper we pursue the Erd\H{o}s--Rothschild versions of Sperner's Theorem, the classic result in extremal set theory on the size of the largest antichain in the Boolean lattice, and Erd\H{o}s' extension to kk-chain-free families. Given a family F\mathcal{F} of subsets of [n][n], we define an (r,k)(r,k)-colouring of F\mathcal{F} to be an rr-colouring of the sets without any monochromatic kk-chains F1F2FkF_1 \subset F_2 \subset \dots \subset F_k. We prove that for nn sufficiently large in terms of kk, the largest kk-chain-free families also maximise the number of (2,k)(2,k)-colourings. We also show that the middle level, ([n]n/2)\binom{[n]}{\lfloor n/2 \rfloor}, maximises the number of (3,2)(3,2)-colourings, and give asymptotic results on the maximum possible number of (r,k)(r,k)-colourings whenever r(k1)r(k-1) is divisible by three.Comment: 30 pages, final versio

    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

    Necessary and Sufficient Conditions on Partial Orders for Modeling Concurrent Computations

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    Partial orders are used extensively for modeling and analyzing concurrent computations. In this paper, we define two properties of partially ordered sets: width-extensibility and interleaving-consistency, and show that a partial order can be a valid state based model: (1) of some synchronous concurrent computation iff it is width-extensible, and (2) of some asynchronous concurrent computation iff it is width-extensible and interleaving-consistent. We also show a duality between the event based and state based models of concurrent computations, and give algorithms to convert models between the two domains. When applied to the problem of checkpointing, our theory leads to a better understanding of some existing results and algorithms in the field. It also leads to efficient detection algorithms for predicates whose evaluation requires knowledge of states from all the processes in the system

    On the Complexity of Mining Itemsets from the Crowd Using Taxonomies

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    We study the problem of frequent itemset mining in domains where data is not recorded in a conventional database but only exists in human knowledge. We provide examples of such scenarios, and present a crowdsourcing model for them. The model uses the crowd as an oracle to find out whether an itemset is frequent or not, and relies on a known taxonomy of the item domain to guide the search for frequent itemsets. In the spirit of data mining with oracles, we analyze the complexity of this problem in terms of (i) crowd complexity, that measures the number of crowd questions required to identify the frequent itemsets; and (ii) computational complexity, that measures the computational effort required to choose the questions. We provide lower and upper complexity bounds in terms of the size and structure of the input taxonomy, as well as the size of a concise description of the output itemsets. We also provide constructive algorithms that achieve the upper bounds, and consider more efficient variants for practical situations.Comment: 18 pages, 2 figures. To be published to ICDT'13. Added missing acknowledgemen
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