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Let V be an n -dimensional vector space over the finite field F q , and let L (V) = ⋃ 0 ≤ k ≤ n [ V k ] 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 = 2 d < n , we determine all optimal and suboptimal s -union antichains completely. For s = 2 d + 1 < n , we prove that an optimal antichain is either [ V d ] or contained in [ V d ] ∪ [ V d + 1 ] and satisfies an equation related to shadows. [ABSTRACT FROM AUTHOR]

Let V be an (n + l) -dimensional vector space over the finite field F q with l ≥ n > 0 and W be a fixed l -dimensional subspace of V. We say that an m -dimensional subspace U of V is of type (m , k) if dim (U ∩ W) = k. Denote the set of all subspaces of type (m , k) in V by M (m , k ; n + l , n). The collection of all the subspaces of types (m , 0) in V , where 0 ≤ m ≤ n , is the attenuated space. In this paper, we prove the Hilton–Milner theorem for M (m , 0 ; n + l , n) , where 3 ≤ m ≤ n. [ABSTRACT FROM AUTHOR]