Given a set $X$ and $k\in\mathbb{N}$ we call a subset of $X$ a $k$-*subset* if its cardinality is $k$. If ${\cal S}$ is a collection of subsets of $X$ and $x\in X$ we set ${\cal S}_x=\{S\in {\cal S}: x\in S\}$.

Let $1<k<\ell$ be integers. Is it possible to find infinitely many integers $n>\ell$ such that there is a collection ${\cal L}$ of $\ell$-subsets of $\{1,\ldots,n\}$ with the following properties?

- every $k$-subset of $\{1,\ldots,n\}$ is contained in exactly one member of ${\cal L}$, and
- for all $a,b\in\{1,\ldots,n\}$ we have $|{\cal L}_a|= |{\cal L}_b|$.

**EDIT.** In the original version of this question I omitted the words "exactly one" in condition 1; thanks to @user44191 for spotting this!

exactlyone $\ell$-subset $\endgroup$ – Dominic van der Zypen Dec 16 '18 at 13:35