Brace–Daykin type inequalities for intersecting families

Let n , k and r ≥ 8 be positive integers. Suppose that a family ℱ ⊂ [ n ] k satisfies F 1 ∩ ⋯ ∩ F r ≠ 0̸ for all F 1 , … , F r ∈ ℱ and ⋂ F ∈ ℱ F = 0̸ . We prove that there exist ϵ r > 0 and n r such that | ℱ | ≤ ( r + 1 ) n − r − 1 k − r + n − r − 1 k − r − 1 holds for all n and k , satisfying n > n r and | k n − 1 2 | < ϵ r .