r-cross t-intersecting families for vector spaces

Fq, and [V Let V be an n-dimensional vector space over the finite field ] denote the family of all k-dimensional subspaces of V . The families F1 C [V k], F2C [ V], . . . , Fr C [V] are k1 k2kr said to be r-cross t-intersecting if dim(F1 n F2 n middot middot middot n Fr) >= t for all Fi E Fi, 1 < i < r. The r-cross t-intersecting families F1, F2, . . . , Fr are said to be non-trivial if dim(n1 <= i <= r nF is an element of Fi F) < 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 = middot middot middot = 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.(c) 2022 Elsevier Inc. All rights reserved.