A product version of the Erdős–Ko–Rado theorem

Let be r-cross t-intersecting, that is, |F1∩⋯∩Fr|⩾t holds for all F1∈F1,…,Fr∈Fr. We prove that for every p,μ∈(0,1) there exists r0 such that for all r>r0, all t with 1⩽t<(1/p−μ)r−1/(1−p)−1, there exist n0 and ϵ so that if n>n0 and |k/n−p|<ϵ, then .