Let $1 \le k \le n$ be integers and let $S_k$ be the set of all $k$-size subsets of $\{1, \ldots, n\}$. I am interested in conditions on $n$ and $k$, such that there exists a permutation $\pi$ of $S_k$ such that the consecutive sets of $S_k$ according to $\pi$ have at least $1$ element in common (including the last and the first set). Also, have such objects been studied, and do they have names?
For example, consider $n = 5$ and $k = 2$. Then, consider the following permutation of $2$-size subsets of $\{1, 2, 3, 4, 5\}$ (where I omit set brackets and commas for simplicity): $$12, 23, 34, 45, 51, 13, 35, 52, 24, 41.$$
However, I don't think there exists such a permutation for $n = 4$ and $k = 2$.