【作者】 王晓雯；

【导师】 王毅；

【作者基本信息】 大连理工大学 ， 基础数学， 2012， 硕士

【摘要】 Erdos-Ko-Rado(简称EKR)定理涉及子集格的交的性质,是研究有限集的交族的最早结果,也是组合极值理论中最经典的结论之一。在过去的几十年间,许多学者研究了EKR定理及其各种推广,给出了EKR定理在其他偏序集上的模拟和极值结构的刻画。本文中我们讨论了EKR定理的一些推广,研究了多重集上的EKR定理。此外,利用所得结果我们给出了极值理论中一些已知结果的简洁证明。全文共分为五章,具体内容概括如下。第一章我们介绍了与EKR定理相关的一些基本结果,包括定理的内容及发展历史。第二章我们着重阐述了EKR定理的若干经典证明方法：移位算子法、投影法、圈序法等其他经典方法。第三章我们研究了EKR定理在反链和多重集上的推广。利用所得结果我们给出了极值理论中一些著名定理的简洁证明。第四章我们讨论了EKR定理在单射集上的模拟。第五章我们介绍了EKR定理的应用和研究方向。更多还原

【Abstract】 The Erdos-Ko-Rado (EKR) theorem, which deals with the intersecting property in the lattice of subset, is the earliest result in researching the intersecting families of a finite set. It is also one of the most famous results in extreme set theory. In the past fifty years, The EKR theorem has been extended, simulated and refined.In this thesis, we present some other extensions of the EKR theorem under certain conditions, and study the theorem on multisets. Moreover, we give simple proofs for several famous theorems in extreme set theory by using our results. The main content of this thesis is arranged as follows.In Chapter1, we present some basic results about the EKR theorem, including background of the theorem.In Chapter2, we focus describing several proving methods of the EKR theorem.In Chapter3, we investigate the extension of the EKR theorem on antichains and multisets. At the same time, we give simple proofs for several famous theorems in extreme set theory by using the obtained results.In Chapter4, we explore simulations of the EKR theorem on the injection collection.In Chapter5, we present some applications of the EKR theorem.更多还原