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