Erd(?)s-Ko-Rado Property Of Posets And Related Topics

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.