2003 Fiscal Year Final Research Report Summary
Research on the minimum ai'ea of convex lattice polygons and multiply intersecting families
| Project/Area Number |
14540131
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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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, 教育学部, 助教授 (00217481)
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| Project Period (FY) |
2002 – 2003
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| Keywords | discrete gepmetry / extremal set theory / lattice point / intersecting family / random walk |
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| Research Abstract |
Let A(n) be the-minimum area of convex Lattice n-gon. We have proved that lim A(n)/n^3 exists, and moreover this value is very close to 0.0185067, which is a little bit less than 1/54 which comes from an obvious construction. This result is accepted by Combinatorica.
A family of subsets is called r-wise t-intersecting if any r edges have at least t common vertices. A family of subsets is called Sperner if it contains no inclusion relationship among subsets. We have determined the maximum size of 4-wise 2-intersecting Sperner families. Then we moved on 3-wise case, which was much more-difficult than 4-wise case. Using random walk method, we finally obtained Erdos-Ko-Rado type inequality for 3-wise 2-intersecting families, and then we succeeded to determine the maximum size of 3-wise 2-intersecting Spernar families. We have wrote two papers on this topic, one appears in J.Comb.Theory, the other has been submitted to the same journal.
I did three lectures about the above results as an invited talk at ZiF research year, opening conference "General theory of information transfer and combinatorics", ISM Symposium "Statistics, Combinatorics and Geometry" and Renyi Institute "Workshop on extremal combinatorics."
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Research Products
(12 results)
