| Budget Amount *help |
¥8,150,000 (Direct Cost: ¥7,700,000、Indirect Cost: ¥450,000)
Fiscal Year 2007: ¥1,950,000 (Direct Cost: ¥1,500,000、Indirect Cost: ¥450,000)
Fiscal Year 2006: ¥2,100,000 (Direct Cost: ¥2,100,000)
Fiscal Year 2005: ¥1,500,000 (Direct Cost: ¥1,500,000)
Fiscal Year 2004: ¥2,600,000 (Direct Cost: ¥2,600,000)
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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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