r-cross t-intersecting families for vector spaces

Let V be an n-dimensional vector space over the finite field Fq, and [Vk] denote family of all k-dimensional subspaces V. The families F1⊆[Vk1],F2⊆[Vk2],…,Fr⊆[Vkr] are said to r-cross t-intersecting if dim⁡(F1∩F2∩⋯∩Fr)≥t for Fi∈Fi,1≤i≤r. F1, F2,…,Fr non-trivial dim⁡(∩1≤i≤r∩F∈FiF)<t. In this paper, we first determine structure with maximum product their sizes. As a consequence, partially prove one Frankl Tokushige's conjectures about 1-intersecting spaces. Then describe sizes under assumptions r=2 F1=F2=⋯=Fr=F, respectively, where F in latter assumption is well known as r-wise family. Meanwhile, stability results also been proved.