Some Research On Combinatorics Of Finite Sets

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.