- First, a combinatorial question fit for an undergrad course. Say I have a collection $\mathcal{C}$ of non-empty subsets of $S=\{1,\dotsc,n\}$ such that every element of $\mathcal{C}$ has at most $k$ elements and every element of $S$ is contained in no fewer than $1$ and no more than $k$ elements of $\mathcal{C}$. Then it is easy to see that there has to be a disjoint subcollection of $\mathcal{C}$ consisting of at least $n/(k^3-k^2+k)$ sets. Is this lower bound optimal?

(To me, the question feels like a discrete analogue of the Vitali covering lemma.)

- Let $A$ be an $n$-by-$n$ matrix whose entries are $0$s, $1$s and $-1$s. Assume that the number of non-zero entries in any row or column is greater than $0$ and no greater than $k$. By 1., the rank of $A$ is at least $n/(k^3-k^2+k)$. Can one give a better lower bound?