Weighted Non-Trivial Multiply Intersecting Families

Let n,r and t be positive integers. A family F of subsets of [n]={1,2, . . . ,n} is called r-wise t-intersecting if |F1∩·· ·∩Fr|≥ t holds for all F1, . . . ,Fr ∈F . An r-wise 1-intersecting family is also called an r-wise intersecting family for short. An r-wise t-intersecting family F is called non-trivial if |⋂F∈F F |<t. Let us define the Brace–Daykin structure as follows. F BD = {F ⊂ [n] : |F ∩ [r + 1]| ≥ r}. Then Fr BD is a non-trivial r-wise intersecting family. Brace and Daykin proved the following.