Weighted Non-Trivial Multiply Intersecting Families

Let n and r be positive integers. Suppose that a family $$ {\user1{\mathcal{F}}} \subset 2^{{{\left[ n \right]}}} $$ satisfies F1∩···∩Fr ≠∅ for all F1, . . .,Fr ∈$$ {\user1{\mathcal{F}}} $$ and $$ {\bigcap {_{{F \in {\user1{\mathcal{F}}}}} } }F = \emptyset $$. We prove that there exists ε=ε(r) >0 such that $$ {\sum {_{{F \in {\user1{\mathcal{F}}}}} } }\omega ^{{{\left| F \right|}}} {\left( {1 - \omega } \right)}^{{n - {\left| F \right|}}} \leqslant \omega ^{r} {\left( {r + 1 - r\omega } \right)} $$ holds for 1/2≤w≤1/2+ε if r≥13.