Weighted Non-Trivial Multiply Intersecting Families

摘要

Let n and r be positive integers. Suppose that a family $ {user1{mathcal{F}}} subset 2^{{{left[ n right]}}} $ satisfies F 1∩···∩F r ≠∅ for all F 1, . . .,F r ∈ $ {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 - romega } right)} $ holds for 1/2≤w≤1/2+ε if r≥13.