SOLVED
Let $k\geq 4$. If $\mathcal{F}$ is a family of subsets of $\{1,\ldots,n\}$ with $\lvert A\rvert=k$ for all $A\in \mathcal{F}$ and $\lvert \mathcal{F}\rvert >\binom{n-2}{k-2}$ then there are $A,B\in\mathcal{F}$ such that $\lvert A\cap B\rvert=1$.
A conjecture of Erdős and Sós. Katona (unpublished) proved this when $k=4$, and Frankl
[Fr77] proved this for all $k\geq 4$.
See also [703].
