Toward a Baranyai Theorem With an Additional Condition

A $\left(k,\ell \right)$ partial partition of an $n$-element set is a collection of $\ell$ pairwise disjoint $k$-element subsets. It is proved that, if $n$ is large enough, one can find $\left(\left(\frac{n}{k}\right)∕\ell \right)$ such partial partitions in such a way that if $A_1$ and $A_2$ are distinct classes in one of the partial partitions, $B_1$ and $B_2$ are distinct classes in another one, then one of the intersections $A_1\cap B_1,A_2\cap B_2$ has size at most $\frac{k}{2}$.