1 research outputs found

    Suboptimal ss-union familes and ss-union antichains for vector spaces

    Full text link
    Let VV be an nn-dimensional vector space over the finite field Fq\mathbb{F}_{q}, and let L(V)=⋃0≀k≀n[Vk]\mathcal{L}(V)=\bigcup_{0\leq k\leq n}\left[V\atop k\right] be the set of all subspaces of VV. A family of subspaces FβŠ†L(V)\mathcal{F}\subseteq \mathcal{L}(V) is ss-union if dim(F+Fβ€²)≀s(F+F')\leq s holds for all FF, Fβ€²βˆˆFF'\in\mathcal{F}. A family FβŠ†L(V)\mathcal{F}\subseteq \mathcal{L}(V) is an antichain if Fβ‰°Fβ€²F\nleq F' holds for any two distinct F,Fβ€²βˆˆFF, F'\in \mathcal{F}. The optimal ss-union families in L(V)\mathcal{L}(V) have been determined by Frankl and Tokushige in 20132013. The upper bound of cardinalities of ss-union (s<n)(s<n) antichains in L(V)\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 ss-union families for vector spaces and then investigates ss-union antichains. For s=ns=n or s=2d<ns=2d<n, we determine all optimal and suboptimal ss-union antichains completely. For s=2d+1<ns=2d+1<n, we prove that an optimal antichain is either [Vd]\left[V\atop d\right] or contained in [Vd]⋃[Vd+1]\left[V\atop d\right]\bigcup \left[V\atop d+1\right] which satisfies an equality related with shadows.Comment: 20 page
    corecore