Weighted Non-Trivial Multiply Intersecting Families

Let n and r be positive integers. Suppose that a family \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\user1{\mathcal{F}}} \subset 2^{{{\left[ n \right]}}} $$\end{document} satisfies F1∩···∩Fr ≠∅ for all F1, . . .,Fr ∈\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\user1{\mathcal{F}}} $$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\bigcap {_{{F \in {\user1{\mathcal{F}}}}} } }F = \emptyset $$\end{document}. We prove that there exists ε=ε(r) >0 such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\sum {_{{F \in {\user1{\mathcal{F}}}}} } }\omega ^{{{\left| F \right|}}} {\left( {1 - \omega } \right)}^{{n - {\left| F \right|}}} \leqslant \omega ^{r} {\left( {r + 1 - r\omega } \right)} $$\end{document} holds for 1/2≤w≤1/2+ε if r≥13.