【作者】 陈娜；

【导师】 阴东升；

【作者基本信息】 北京工业大学 ， 数学， 2020， 硕士

【摘要】 我们研究有限集的交族问题:当对[n]元集([n]={1,2,3,...,n})的子集族的交、对子集族中的子集所含元素个数进行限制或者将两者同时进行限制时,考虑该子集族所含子集个数的上界。也就是研究满足所给条件的子集族的最大基数问题。本文结合有限集交族中现有的一些组合对象和组合结论,利用多项式空间的方法,研究非模情形和模p情形下的特定的有限集L-交族以及特定的交叉交族问题,并且在Hunter Snevily,陈永川,刘九强以及Rudy XJ Liu等人的相关研究结果的基础上做进一步的推广。首先,用多项式空间的方法对模pL-交的子集族F分别满足:(1)K∩L=(?),∩Fi∈F Fi≠(?),min ki＞s-r;(2)min ki＞max lj,n ≥s+max ki时的最大基数给出详细证明。其次,用该方法证明并得出关于特定交叉交族的两个结论:(1)模p情形下,满足min{|Ai|(modp)|1 ≤i ≤m}>max lj,n≥s+maxki时,子集族A或B所含子集的最大基数;(2)非模情形下,满足{|Ai||1≤i≤m}为r个连续整数,k1>s-r,max lj＜min{|Ai||1 ≤i≤m}时,子集族A或B所含子集的最大基数。最后,给出非模情形下,L-交的子集族F在满足:min ki＞max lj,∩Fi∈FFi≠(?)时的最大基数,得到定理4.2,并利用定理4.1的结论对定理4.2进行证明。更多还原

【Abstract】 We study the problem of intersection families of finite sets,that is,when we re-strict the intersection of subset families of n-ary sets([n]={1,2,3,…,n}),or when the number of elements contained in a subset is restricted,or both of them are restricted at the same time,we observe the upper bound of the number of subsets contained in the subset family.In other words,the problem of the largest cardinality of the subset family that satisfies the given conditions is studied.This paper combines the existing combi-nation objects and combination conclusions in the intersection family of finite sets,and uses the method of polynomial space to study the specific L-intersecting families and the cross-L-intersecting families in the modular p version and a non-modular version respectively.Based on the results obtained by Hunter Snevily,Chen Yongchuan,Liu Jiuqiang,and Rudy XJ Liu,etc.,we will make further promotion.Firstly,we use the method of polynomial space to prove the maximum cardinality when the subset family satisfies the following two conditions respectively in the case of module p:(1)K∩L=(?),∩Fi∈FFi≠(?)and min ki>s-r;(2)min ki＞max lj and n≥s+max ki;Secondly,this method is used to prove the maximum cardinality of the spe-cific cross-L-intersecting families:(1)in the modular p version,min{|Ai|(modp)|1≤i ≤m}＞ max lj,n≥ s+-max ki;(2)in a non-modular version,where {|Ai||1≤i≤m}is r consecutive integers,k1＞ s-r andmax lj ＜min{|Ai||1≤i≤m}.Finally,we study the maximum cardinality when the subset family satisfies:min ki>max lj and∩Fi∈FFi≠(?)in the non-modular version,then the Theorem 4.2 is obtained and we use the conclusion of Theorem 4.1 to prove Theorem 4.2.更多还原