Non-trivial T-Intersecting Separated Families

Let n,k,ℓ,t be positive integers with k≥ℓ≥t+2 and let X=X1⊎X2⊎⋯⊎Xk, |Xi|=n. A family F of ℓ-subsets of X is called a separated family if |F∩Xi|≤1 for all F∈F and i=1,2,…,k. A separated family F is called non-trivial t-intersecting if |F∩F′|≥t for all F,F′∈F and |∩{F:F∈F}|(t+1)(ℓ−t−1)2(k−t−1)+1.