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Suboptimal -union familes and -union antichains for vector spaces
Let be an -dimensional vector space over the finite field
, and let be the set of all subspaces of . A family of subspaces
is -union if dim holds
for all , . A family
is an antichain if holds for any two distinct . The optimal -union families in have been
determined by Frankl and Tokushige in . The upper bound of cardinalities
of -union antichains in has been established by
Frankl recently, while the structures of optimal ones have not been displayed.
The present paper determines all suboptimal -union families for vector
spaces and then investigates -union antichains. For or , we
determine all optimal and suboptimal -union antichains completely. For
, we prove that an optimal antichain is either or contained in which satisfies an equality related with shadows.Comment: 20 page