An improved threshold for the number of distinct intersections of intersecting families

A family $\mathcal{F}$ of subsets of $\{1,2,\dots ,n\}$ is called a $t$-intersecting family if $\mathrm{\mid }F\cap G\mathrm{\mid }\ge t$ 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 $\mathcal{I}(\mathcal{F})=\{F\cap G:F,G\in \mathcal{F}\text{ and }F\mathrm{\ne }G\}$ 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 $\{1,2,\dots ,n\}$. Their theorem holds for integers $n\ge 50k^2$. 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 $k^{3\mathrm{/}2+o(1)}$ for the intersecting families.