2007 Fiscal Year Final Research Report Summary
Research on extremal structures in combinatorics
| Project/Area Number |
16340027
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
General mathematics (including Probability theory/Statistical mathematics)
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| Research Institution | University of the Ryukyus |
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Principal Investigator |
TOKUSHIGE Norihide University of the Ryukyus, College of Education, Associate Professor (00217481)
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| Project Period (FY) |
2004 – 2007
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| Keywords | Extremal set theory / multiply intersecting families / random walk / Szemeredi regularity lemma / regularity method / packing / Erdos-ko-Rado theorem |
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| Research Abstract |
(1) We study extremal structures of multiply intersecting families. We developed the random walk method introduced by P. Frankl. One of our new ideas is to associate weighted size (p-weight) of non-uniform hypergraphs with k-uniform hypergraphs. Here p and k/n are corresponding, where n is the number of vertices of hypergraphs. We determined the maximal size of r-wise t-intersecting k-uniform hypergraphs, which is a generalization of the Erdos-Ko-Rado theorem. We also determined the maximal size of nontrivial t-intersecting families and t-intersecting Sperner families. These were based on a joint work with P. Frankl.
(2) We gave alternative proofs of density version of some combinatorial partition theorems originally obtained by Szemeredi, Furstenberg and Katznelson. This was a joing work with V. Rodl, M. Schacht, E. Tengan. Our proofs are based on an extremal hypergraph result which was independently obtained by Gowers and Nagle-Rodl-Schacht-Skokan by extending Szemeredi's regularity lemma to hypergraph.
(3) The problem of finding the integer packing number of a k-uniform hypergraph is an NP-hard problem. Find the fractinal packing number however can be done in polynomial time. We gave a lower bound for the integer packing number in terms of the fractional packing number.
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[Presentation] 乱歩と交差族2006
Author(s)
徳重典英
Organizer
応用数学合同研究集会
Place of Presentation
龍谷大学
Year and Date
2006-12-20
Description
「研究成果報告書概要(和文)」より
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[Presentation] An extension of the Erdos-Ko-Rado theorem2006
Author(s)
N. Tokushige
Organizer
Sixth Czech-Slovak international symposium. Combinatorics, graph theory, algorithms and applications
Place of Presentation
Technical University, Prague, Czech
Year and Date
2006-07-10
Description
「研究成果報告書概要(和文)」より
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[Presentation] 極値集合論の話2006
Author(s)
徳重典英
Organizer
第2回組合せ論若手研究集会
Place of Presentation
慶應義塾大学
Year and Date
2006-02-20
Description
「研究成果報告書概要(和文)」より
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