Suboptimal s-union families and s-union antichains for vector spaces

Abstract

Let V be an n-dimensional vector space over the finite field Fq, and let L(V)=⋃0≤k≤n[Vk] be the set of all subspaces of V. A family of subspaces F⊆L(V) is s-union if dim(F+F′)≤s holds for all F, F′∈F. A family F⊆L(V) is an antichain if F≰F′ holds for any two distinct F,F′∈F. An s-union family (resp. s-union antichain) with the largest possible size is called optimal. The optimal s-union families in L(V) have been determined by Frankl and Tokushige in 2013. An upper bound on cardinalities of s-union (s<n) antichains in L(V) has been established by Frankl recently, while the structures of optimal ones have not been displayed. The largest s-union families (resp. s-union antichains) that are not contained in any optimal one are called suboptimal. The present paper determines all suboptimal s-union families for vector spaces and then investigates s-union antichains. For s=n or s=2d<n, we determine all optimal and suboptimal s-union antichains completely. For s=2d+1<n, we prove that an optimal antichain is either [Vd] or contained in [Vd]∪[Vd+1] and satisfies an equation related to shadows.