The Lower Shadow And The Scattered System In The Linear Space On The Binary Domain

Kruskal-Katona theorem is an important and classical theorem in extremal combi-natorics theory for finite sets,which has found many applications in combinatorics.It determined the minimum size of the shadow of a collection of m sets of size k on a finite set.In this paper,we explore the similar problem of shadow size minimization in linear space,study the shadow size of k-dimensional subspace family in linear space over binary field F2,introduce the concept of scattered intersecting family,which is closely related to the shadow size of sub space family in linear space,and use the method of linear algebra to get the maximum size of scattered intersecting family,so as to finally determine the min-imum shadow size of k-dimensional subspace families whose size is less than 2k,and for k-dimensional subspace families whose size is more than 2k,a non-strict lower bound of its shadow size is given.In the end of this paper,the structure of the subspace family with the minimum shadow size,i.e.the scattered intersecting family,is analyzed in detail,thus the structure of the scattered intersecting family is given,and the number of the maximum scattered intersecting family is counted clearly.