Best possible bounds on the number of distinct differences in intersecting families

For a family $mathcal F$, let $mathcal D(mathcal F)$ stand for the family of all sets that can be expressed as $Fsetminus G$, where $F,Gin mathcal F$. A family $mathcal F$ is intersecting if any two sets from the family have non-empty intersection. In this paper, we study the following question: what is the maximum of $|mathcal D(mathcal F)|$ for an intersecting family of $k$-element sets? Frankl conjectured that the maximum is attained when $mathcal F$ is the family of all sets containing a fixed element. We show that this holds if $n>50klog k$ and $k>50$. At the same time, we provide a counterexample for $n< 4k.$