Brace-Daykin type inequalities for intersecting families

Let n,k and r≥8 be positive integers. Suppose that a family ℱ⊂[n]k satisfies F1∩⋯∩Fr≠0̸ for all F1,…,Fr∈ℱ and ⋂F∈ℱF=0̸. We prove that there exist ϵr>0 and nr such that |ℱ|≤(r+1)n−r−1k−r+n−r−1k−r−1 holds for all n and k, satisfying n>nr and |kn−12|<ϵr.