225 research outputs found

    Ramsey numbers for partially-ordered sets

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    We present a refinement of Ramsey numbers by considering graphs with a partial ordering on their vertices. This is a natural extension of the ordered Ramsey numbers. We formalize situations in which we can use arbitrary families of partially-ordered sets to form host graphs for Ramsey problems. We explore connections to well studied Tur\'an-type problems in partially-ordered sets, particularly those in the Boolean lattice. We find a strong difference between Ramsey numbers on the Boolean lattice and ordered Ramsey numbers when the partial ordering on the graphs have large antichains.Comment: 18 pages, 3 figures, 1 tabl

    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

    Multiplicity of the trivial representation in rank-selected homology of the partition lattice

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    We study the multiplicity bS(n)b_S(n) of the trivial representation in the symmetric group representations Ξ²S\beta_S on the (top) homology of the rank-selected partition lattice Ξ nS\Pi_n^S. We break the possible rank sets SS into three cases: (1) 1∉S1\not\in S, (2) S=1,...,iS=1,..., i for iβ‰₯1i\ge 1 and (3) S=1,...,i,j1,...,jlS=1,..., i,j_1,..., j_l for i,lβ‰₯1i,l\ge 1, j1>i+1j_1 > i+1. It was previously shown by Hanlon that bS(n)=0b_S(n)=0 for S=1,...,iS=1,..., i. We use a partitioning for Ξ”(Ξ n)/Sn\Delta(\Pi_n)/S_n due to Hersh to confirm a conjecture of Sundaram that bS(n)>0b_S(n)>0 for 1∉S1\not\in S. On the other hand, we use the spectral sequence of a filtered complex to show bS(n)=0b_S(n)=0 for S=1,...,i,j1,...,jlS=1,..., i,j_1,..., j_l unless a certain type of chain of support SS exists. The partitioning for Ξ”(Ξ n)/Sn\Delta(\Pi_n)/S_n allows us then to show that a large class of rank sets S=1,...,i,j1,...,jlS=1,..., i,j_1,..., j_l for which such a chain exists do satisfy bS(n)>0b_S(n)>0. We also generalize the partitioning for Ξ”(Ξ n)/Sn\Delta(\Pi_n)/S_n to Ξ”(Ξ n)/SΞ»\Delta(\Pi_n)/S_{\lambda}; when Ξ»=(nβˆ’1,1)\lambda = (n-1,1), this partitioning leads to a proof of a conjecture of Sundaram about S1Γ—Snβˆ’1S_1\times S_{n-1}-representations on the homology of the partition lattice

    Almost partitioning the hypercube into copies of a graph

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    Let H be an induced subgraph of the hypercube Qk, for some k. We show that for some c=c(H), the vertices of Qn can be partitioned into induced copies of H and a remainder of at most O(nc) vertices. We also show that the error term cannot be replaced by anything smaller than log

    Almost partitioning the hypercube into copies of a graph

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    Let H be an induced subgraph of the hypercube Qk, for some k. We show that for some c=c(H), the vertices of Qn can be partitioned into induced copies of H and a remainder of at most O(nc) vertices. We also show that the error term cannot be replaced by anything smaller than log
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