Non-trivial r-wise intersecting families

A k-uniform family F⊂[n]k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{F} \subset \binom{[n]}{k}$$\end{document}is called non-trivial r-wise intersecting if F1∩F2∩⋯∩Fr≠∅\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_1{\cap} F_{2} {\cap}{\cdots}{\cap} F_{r} \neq \emptyset$$\end{document} for every F1,F2,…,Fr∈F\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_{1}, F_{2},{\ldots},F_{r} {\in} \mathcal{F}$$\end{document} and ∩F=∅\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\cap \mathcal{F} = \emptyset$$\end{document}. O’Neill and Verstraëte determined the maximum size of a non-trivial r-wise intersecting family for n sufficiently large. Actually, the Hilton–Milner–Frankl Theorem implies O’Neill–Verstraëte's result for n≥r(k-r+2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n \geq r(k-r + 2)$$\end{document}. In the present paper, we show that the same result holds for a certain range when n is close to 2k.