A multiply intersecting Erdős–Ko–Rado theorem — The principal case

Let m ( n , k , r , t ) be the maximum size of F ⊂ [ n ] k satisfying | F 1 ∩ ⋯ ∩ F r | ≥ t for all F 1 , … , F r ∈ F . We prove that for every p ∈ ( 0 , 1 ) there is some r 0 such that, for all r > r 0 and all t with 1 ≤ t ≤ ⌊ ( p 1 − r − p ) / ( 1 − p ) ⌋ − r , there exists n 0 so that if n > n 0 and p = k / n , then m ( n , k , r , t ) = n − t k − t . The upper bound for t is tight for fixed p and r .