Inverse problems of the Erdos-Ko-Rado type theorems for families of vector spaces and permutations  

作　　者：Xiangliang Kong Yuanxiao Xi Bingchen Qian Gennian Ge 

机构地区：[1]School of Mathematical Sciences,Capital Normal University,Beijing 100048,China [2]School of Mathematical Sciences,Zhejiang University,Hangzhou 310027,China

出　　处：《Science China Mathematics》2022年第5期1081-1108,共28页中国科学：数学（英文版）

基　　金：supported by National Natural Science Foundation of China (Grant No. 11971325);National Key Research and Development Program of China (Grant Nos. 2020YFA0712100 and 2018YFA0704703);Beijing Scholars Program

摘　　要：Ever since the famous Erd os-Ko-Rado theorem initiated the study of intersecting families of subsets,extremal problems regarding intersecting properties of families of various combinatorial objects have been extensively investigated.Among them,studies about families of subsets,vector spaces and permutations are of particular concerns.Recently,we proposed a new quantitative intersection problem for families of subsets:For F([n]k),define its total intersection number as I(F)=ΣF1;F2∈F|F1∩F2|.Then,what is the structure of F when it has the maximal total intersection number among all the families in([n]k)with the same family size?In a recent paper,Kong and Ge(2020)studied this problem and characterized extremal structures of families maximizing the total intersection number of given sizes.In this paper,we consider the analogues of this problem for families of vector spaces and permutations.For certain ranges of family sizes,we provide structural characterizations for both families of subspaces and families of permutations having maximal total intersection numbers.To some extent,these results determine the unique structure of the optimal family for some certain values of jFj and characterize the relationship between having the maximal total intersection number and being intersecting.Besides,we also show several upper bounds on the total intersection numbers for both families of subspaces and families of permutations of given sizes.

关 键 词：total intersection number vector spaces PERMUTATIONS 

分 类 号：O144[理学—数学]