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LetX be ann-element set and letA and? be families of subsets ofX. We say thatA and? are crosst-intersecting if |A ∩ B| ≥ t holds for all A ∈A and for allB ∈ ?. Suppose thatA and ? are crosst-intersecting. This paper first proves a crosst-intersecting version of Harper's Theorem:
- There are two crosst-intersecting Hamming spheresA 0,? 0 with centerX such that |A| ≤ |A 0| and|?| ≤ |? 0| hold.
- Suppose thatt ≥ 2 and that the pair of integers (|A) is maximal with respect to direct product ordering among pairs of crosst-intersecting families. Then,A and? are Hamming spheres with centerX.
- Ifn + t = 2k ? 1 then |A| |?| ≤ max \(\left\{ {\left( {K_k^n + \left( {_{k - 1}^{n - 1} } \right)} \right)^2 ,K_k^n K_{k - 1}^n } \right\}\) holds, whereK l n is defined as \(\left( {_n^n } \right)\left( {_{n - 1}^n } \right) + \cdots + \left( {_l^n } \right).\)
- Ifn + t = 2k then |A| |? ≤ (K k n )2 holds.
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Let be an n-uniform hypergraph on 2n vertices. Suppose that and holds for all F1,F2,F3 ∈ . We prove that the size of is at most .
The second author was supported by MEXT Grant-in-Aid for Scientific Research (B) 16340027 相似文献
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Let n and r be positive integers. Suppose that a family
satisfies F1∩···∩Fr ≠∅ for all F1, . . .,Fr ∈
and
. We prove that there exists ε=ε(r) >0 such that
holds for 1/2≤w≤1/2+ε if r≥13. 相似文献
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Hiroshi Maehara Katsuhiro Ota Norihide Tokushige 《Journal of Combinatorial Theory, Series A》1997,80(2):290-294
We prove that every finite simple graph can be drawn in the plane so that any two vertices have an integral distance if and only if they are adjacent. The proof is constructive. 相似文献
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Norihide Tokushige 《组合设计杂志》2006,14(1):52-55
Let be a k‐uniform hypergraph on n vertices. Suppose that holds for all . We prove that the size of is at most if satisfies and n is sufficiently large. © 2005 Wiley Periodicals, Inc. J Combin Designs 相似文献
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Hiroshi Suzuki Tetsu Itotagawa Yuli Setyo Indartono Hiromoto Usui Norihide Wada 《Rheologica Acta》2006,46(2):287-295
Rheological characteristics of trimethylolethane (TME) clathrate–hydrate slurry treated with drag-reducing surfactants were investigated. Friction coefficients and apparent viscosities were measured when the concentration of TME and its hydrate fraction treated with and without drag-reducing surfactants were changed in several steps. From the results, it is found that the surfactant addition causes effective drag reduction in a pipe flow when the hydrate fraction becomes high, while effective drag reduction disappears in the cases of low hydrate fraction. The results of viscosity measurements indicate that the TME molecules disturb the formation of shear-induced structures (SIS) causing drag reduction phenomena. To investigate this interaction between TME and surfactant micelles, the effect of TME concentration on viscosity and relaxation time of solutions was discussed. From this, it was found out that there exists a critical concentration of TME on the formation of SIS and that it becomes larger as shear rate increases. Thus, we conclude that this interaction between TME and micellar structures causes less drag reduction for the cases of low hydrate fraction, while the drag reduction appears in cases of high hydrate fraction because TME concentration in liquid phase becomes small. 相似文献