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178
The combinatorics of splittability
, 2004
"... Marion Scheepers, in his studies of the combinatorics of open covers, introduced the property Split(U; V) asserting that a cover of type U can be split into two covers of type V. In the first part of this paper we give an almost complete classification of all properties of this form where U and V ar ..."
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Cited by 2 (1 self)
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Marion Scheepers, in his studies of the combinatorics of open covers, introduced the property Split(U; V) asserting that a cover of type U can be split into two covers of type V. In the first part of this paper we give an almost complete classification of all properties of this form where U and V
On the splittability of infinite covers
, 2007
"... Let X be a set, κ be a cardinal number and let H a family of subsets of X which covers each x ∈ X at least κ times. Under what assumptions can we decompose H into κ many disjoint subcovers? We examine this problem under various assumptions on the set X and on the cover H: among other situations we c ..."
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Let X be a set, κ be a cardinal number and let H a family of subsets of X which covers each x ∈ X at least κ times. Under what assumptions can we decompose H into κ many disjoint subcovers? We examine this problem under various assumptions on the set X and on the cover H: among other situations we consider covers of topological spaces by closed sets, interval covers of linearly ordered sets and covers of R n by polyhedra and by arbitrary convex sets. We focus mainly on these problems when κ is an infinite cardinal. Besides numerous positive and negative results many questions turn out to be independent of the axioms of set theory. 1
SPLITTABLE IDEALS AND THE RESOLUTIONS OF MONOMIAL IDEALS
, 2006
"... We provide a new combinatorial approach to study the minimal free resolutions of edge ideals, that is, quadratic squarefree monomial ideals. With this method we can recover most of the known results on resolutions of edge ideals with fuller generality, and at the same time, obtain new results. Pa ..."
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Cited by 36 (6 self)
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We provide a new combinatorial approach to study the minimal free resolutions of edge ideals, that is, quadratic squarefree monomial ideals. With this method we can recover most of the known results on resolutions of edge ideals with fuller generality, and at the same time, obtain new results. Past investigations on the resolutions of edge ideals usually reduced the problem to computing the dimensions of reduced homology or Koszul homology groups. Our approach circumvents the highly nontrivial problem of computing the dimensions of these groups and turns the problem into combinatorial questions about the associated simple graph. We also show that our technique extends successfully to the study of graded Betti numbers of arbitrary squarefree monomial ideals viewed as facet ideals of simplicial complexes.
Splittings and Ramsey properties of permutation classes
 arXiv:1307.0027 [math.CO]. Cited on
"... We say that a permutation pi is merged from permutations ρ and τ, if we can color the elements of pi red and blue so that the red elements are orderisomorphic to ρ and the blue ones to τ. A permutation class is a set of permutations closed under taking subpermutations. A permutation class C is spli ..."
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Cited by 1 (0 self)
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is splittable if it has two proper subclasses A and B such that every element of C can be obtained by merging an element of A with an element of B. Several recent papers use splittability as a tool in deriving enumerative results for specific permutation classes. The goal of this paper is to study splittability
SPLITTINGS OF MONOMIAL IDEALS
, 2008
"... We provide some new conditions under which the graded Betti numbers of a monomial ideal can be computed in terms of the graded Betti numbers of smaller ideals, thus complementing Eliahou and Kervaire’s splitting approach. As applications, we show that edge ideals of graphs are splittable, and we pr ..."
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Cited by 12 (0 self)
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We provide some new conditions under which the graded Betti numbers of a monomial ideal can be computed in terms of the graded Betti numbers of smaller ideals, thus complementing Eliahou and Kervaire’s splitting approach. As applications, we show that edge ideals of graphs are splittable, and we
Homogeneous representations of KhovanovLauda algebras
 J. EUROP. MATH. SOC
, 2008
"... We construct irreducible graded representations of simply laced KhovanovLauda algebras which are concentrated in one degree. The underlying combinatorics of skew shapes and standard tableaux corresponding to arbitrary simply laced types has been developed previously by Peterson, Proctor and Stemb ..."
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Cited by 21 (8 self)
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We construct irreducible graded representations of simply laced KhovanovLauda algebras which are concentrated in one degree. The underlying combinatorics of skew shapes and standard tableaux corresponding to arbitrary simply laced types has been developed previously by Peterson, Proctor
Results 1  10
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178