225 research outputs found
Ramsey numbers for partially-ordered sets
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
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
We study the multiplicity of the trivial representation in the
symmetric group representations on the (top) homology of the
rank-selected partition lattice . We break the possible rank sets
into three cases: (1) , (2) for and (3)
for , . It was previously shown
by Hanlon that for . We use a partitioning for
due to Hersh to confirm a conjecture of Sundaram that
for . On the other hand, we use the spectral sequence of
a filtered complex to show for unless a
certain type of chain of support exists. The partitioning for
allows us then to show that a large class of rank sets
for which such a chain exists do satisfy .
We also generalize the partitioning for to
; when , this partitioning leads
to a proof of a conjecture of Sundaram about -representations on the homology of the partition lattice
Almost partitioning the hypercube into copies of a graph
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
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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