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偏序集的Erd（？）s-Ko-Rado性质及相关问题
Erd(?)s-Ko-Rado Property of Posets and Related Topics
【作者】 张俊；
【导师】 王军；
【作者基本信息】 大连理工大学 ， 基础数学， 2008， 博士
【摘要】 论文主要研究偏序集的Erd(?)s-Ko-Rado(EKR)性质,交族的正规匹配(NM)性质、LYM性质,子空间族的最小下影问题以及Erd(?)s-Ko-Rado定理在置换群中的模拟。本文分四章。第一童介绍EKR定理的内容、发展历史以及几种经典证明方法。第二章研究分次偏序集中交族的正规匹配(NM)性质和LYM性质。在较广的意义下我们给出Sperner型性质和交性质的一个统一的处理,于是就自然地引入了分次偏序集中交族的NM性质和LYM性质,证明交-NM性质蕴含交-LYM性质(反之不然)。接着用移位算子证明了B_n的严格交-NM性质。第三章研究子空间族的最小下影问题。用群作用的方法得到L_n(q)到B_n的保秩且保序的映射,这样就可以把L_n(q)看作是加权的B_n。由此得到一个相应的Kruskal-Katona定理,它可以看作是加权的Kruskal-Katona定理。第四章研究对称群和Coxetcr群的EKR性质,通过计算不动点,我们给出对称群的严格EKR性质的一个简单证明,在此基础上证明了Coxeter群的严格EKR性质。
【Abstract】 This thesis investigates the Erd(o|¨)s-Ko-Rado(EKR) property of intersecting families, the normalized matching(NM) property and the LYM property of intersecting antichains. the shadow minimization problem of subspace families,and analogs of Erd(o|¨)s-Ko-Rado Theorem in permutation groups.The thesis consists of four chapters.The first contributes to the contents,the history, and the classic proofs of Erd(o|¨)s-Ko-Rado Theorem.The second devotes to the NM property and the LYM property of intersecting antichains.In a more general setting we give a unified treatment of the Sperner-type properties and the intersecting property in a ranked poset.Then,the LYM property and the(strict) normalized matching(NM) property for intersecting families in a ranked poser are introduced in a natural way.We prove that the NM property implies the LYM property for intersecting families,but the converse is not generally true.By the shift operation we establish the strict NM property for intersecting families in the boolean lattices.The third investigates the shadow minimization problem of subspace families.By the group action method,we obtain a rank-preserving and order-preserving map from L_n(q) to B_n,so that L_n(q) can be considered as weighted B_n,thus we yield a corresponding Kruskal-Katona Theorem.which can be considered as a weighted Kruskal-Katona Theorem.The fourth considers the EKR property of symmetric groups and Coxeter groups.By analyzing the fixed points of permutations,we present a short proof of EKR property of symmetric groups.Based on this we establish EKR-type theorems for Coxeter groups of types B and D.