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Venn Diagrams and Symmetric Chain Decompositions in the Boolean Lattice
 Electron. J. Combin., 11:Research Paper
, 2004
"... We show that symmetric Venn diagrams for n sets exist for every prime n, settling an open question. Until this time, n = 11 was the largest prime for which the existence of such diagrams had been proven. We show that the problem can be reduced to finding a symmetric chain decomposition, satisfying a ..."
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Cited by 15 (2 self)
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We show that symmetric Venn diagrams for n sets exist for every prime n, settling an open question. Until this time, n = 11 was the largest prime for which the existence of such diagrams had been proven. We show that the problem can be reduced to finding a symmetric chain decomposition, satisfying a certain cover property, in a subposet of the Boolean lattice Bn , and prove that such decompositions exist for all prime n.
TRACES OF FINITE SETS: EXTREMAL PROBLEMS AND GEOMETRIC APPLICATIONS
, 1992
"... Given a hypergraph H and a subset S of its vertices, the trace of H on S is defined as HS = {E ∩ S: E ∈ H}. The Vapnik–Chervonenkis dimension (VCdimension) of H is the size of the largest subset S for which HS has 2 S edges. Hypergraphs of small VCdimension play a central role in many areas o ..."
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Cited by 13 (0 self)
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Given a hypergraph H and a subset S of its vertices, the trace of H on S is defined as HS = {E ∩ S: E ∈ H}. The Vapnik–Chervonenkis dimension (VCdimension) of H is the size of the largest subset S for which HS has 2 S edges. Hypergraphs of small VCdimension play a central role in many areas of statistics, discrete and computational geometry, and learning theory. We survey some of the most important results related to this concept with special emphasis on (a) hypergraph theoretic methods and (b) geometric applications.
Saturated chain partitions in ranked partially ordered sets, and nonmonotone symmetric 11Venn diagrams
 Studia Scientiarum Mathematicarum Hungarica
"... In this paper we show that there are at least 2 110 nonisomorphic 11doilies, that is, there are many nonisomorphic symmetric, nonsimple, nonmonotone 11Venn diagrams, with “many ” vertices. We do not achieve the maximum vertex set size, 2046, but we approach it closely, improving from the previ ..."
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Cited by 1 (0 self)
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In this paper we show that there are at least 2 110 nonisomorphic 11doilies, that is, there are many nonisomorphic symmetric, nonsimple, nonmonotone 11Venn diagrams, with “many ” vertices. We do not achieve the maximum vertex set size, 2046, but we approach it closely, improving from the previous 462 in [10] to 1837. The doilies constructed here cannot be constructed by either of the methods of [10] or [6]. The main purpose of this paper is not to publish these attractive diagrams but to inspire new studies by raising ideas, methods, questions, and conjectures, hoping for results analogous to those generated in [10]. These ideas connect two seemingly distant areas of mathematics: a special area of combinatorial geometry, namely, certain families of simple closed Jordan curves in the plane, and the study of ranked partially ordered sets or posets. 1