A product version of the Erdős–Ko–Rado theorem

Let F 1 , … , F r ⊂ ( [ n ] k ) be r-cross t-intersecting, that is, | F 1 ∩ ⋯ ∩ F r | ⩾ t holds for all F 1 ∈ F 1 , … , F r ∈ F r . We prove that for every p , μ ∈ ( 0 , 1 ) there exists r 0 such that for all r > r 0 , all t with 1 ⩽ t < ( 1 / p − μ ) r − 1 / ( 1 − p ) − 1 , there exist n 0 and ϵ so that if n > n 0 and | k / n − p | < ϵ , then | F 1 | ⋯ | F r | ⩽ ( n − t k − t ) r .