Suboptimal $s$-union familes and $s$-union antichains for vector spaces

Let $V$ be an $n$-dimensional vector space over the finite field $\mathbb{F}_{q}$, and let $\mathcal{L}(V)=\bigcup_{0\leq k\leq n}\left[V\atop k\right]$ be the set of all subspaces of $V$. A family of subspaces $\mathcal{F}\subseteq \mathcal{L}(V)$ is $s$-union if dim$(F+F')\leq s$ holds for all $F$, $F'\in\mathcal{F}$. A family $\mathcal{F}\subseteq \mathcal{L}(V)$ is an antichain if $F\nleq F'$ holds for any two distinct $F, F'\in \mathcal{F}$. The optimal $s$-union families in $\mathcal{L}(V)$ have been determined by Frankl and Tokushige in $2013$. The upper bound of cardinalities of $s$-union $(s<n)$ antichains in $\mathcal{L}(V)$ has been established by Frankl recently, while the structures of optimal ones have not been displayed. 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 $\left[V\atop d\right]$ or contained in $\left[V\atop d\right]\bigcup \left[V\atop d+1\right]$ which satisfies an equality related with shadows.

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