A family F\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {F}$$\end{document} of subsets of {1,2,…,n}\documentclass[12pt]{minimal}
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\begin{document}$$\{1,2,\ldots ,n\}$$\end{document} is called a t-intersecting family if |F∩G|≥t\documentclass[12pt]{minimal}
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\begin{document}$$|F\cap G| \ge t$$\end{document} for any two members F,G∈F\documentclass[12pt]{minimal}
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\begin{document}$$F, G \in \mathcal {F}$$\end{document} and for some positive integer t. If t=1\documentclass[12pt]{minimal}
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\begin{document}$$t=1$$\end{document}, then we call the family F\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {F}$$\end{document} to be intersecting. Define the set I(F)={F∩G:F,G∈FandF≠G}\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {I}(\mathcal {F}) = \{F\cap G: F, G \in \mathcal {F} \text { and } F \ne G\}$$\end{document} to be the collection of all distinct intersections of F\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {F}$$\end{document}. Frankl et al. proved an upper bound for the size of I(F)\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {I}(\mathcal {F})$$\end{document} of intersecting families F\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {F}$$\end{document} of k-subsets of {1,2,…,n}\documentclass[12pt]{minimal}
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\begin{document}$$\{1,2,\ldots ,n\}$$\end{document}. Their theorem holds for integers n≥50k2\documentclass[12pt]{minimal}
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\begin{document}$$n \ge 50 k^2$$\end{document}. In this article, we prove an upper bound for the size of I(F)\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {I}(\mathcal {F})$$\end{document} of t-intersecting families F\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {F}$$\end{document}, provided that n exceeds a certain number f(k, t). Along the way we also improve the threshold k2\documentclass[12pt]{minimal}
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\begin{document}$$k^2$$\end{document} to k3/2+o(1)\documentclass[12pt]{minimal}
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\begin{document}$$k^{3/2+o(1)}$$\end{document} for the intersecting families.
