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

Let m(n,k,r,t)m(n,k,r,t) be the maximum size of F⊂[n]k satisfying |F1∩⋯∩Fr|≥t|F1∩⋯∩Fr|≥t for all F1,…,Fr∈FF1,…,Fr∈F. We prove that for every p∈(0,1)p∈(0,1) there is some r0r0 such that, for all r>r0r>r0 and all tt with 1≤t≤⌊(p1−r−p)/(1−p)⌋−r1≤t≤⌊(p1−r−p)/(1−p)⌋−r, there exists n0n0 so that if n>n0n>n0 and p=k/np=k/n, then m(n,k,r,t)=n−tk−t. The upper bound for tt is tight for fixed pp and rr.