Cross $t$-intersecting families for symplectic polar spaces

Abstract

Let P be a symplectic polar space over a finite field F q , and P m denote the collection of all k-dimensional totally isotropic subspace in P. Let F 1 ⊂ P m1 andt for any F 1 ∈ F 1 and F 2 ∈ F 2 . We say they are cross tintersecting families. Moreover, we say they are trivial if each member of them contains a fixed t-dimensional totally isotropic subspace. In this paper, we show that cross tintersecting families with maximum product of sizes are trivial. We also describe the structure of non-trivial t-intersecting families with maximum product of sizes.