R-cross t-intersecting families for vector spaces

Abstract

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