  For a $k$-uniform hypergraph $H$, let $\nu^{(m)}(H)$ denote the maximum size
of a set $S$ of edges of $H$ whose pairwise intersection has size less than
$m$. Let $\tau^{(m)}(H)$ denote the minimum size of a set $S$ of $m$-sets of
$V(H)$ such that every edge of $H$ contains some $m$-set from $S$. A conjecture
by Aharoni and Zerbib, which generalizes a conjecture of Tuza on the size of
minimum edge covers of triangles of a graph, states that for a $k$-uniform
hypergraph $H$, $\tau^{(k - 1)}(H)/\nu^{(k - 1)}(H) \leq \left \lceil \frac{k +
1}{2} \right \rceil$. In this paper, we show that this generalization of Tuza's
conjecture holds when $\nu^{(k - 1)}(H) \leq 3$. As a corollary, we obtain a
graph class which satisfies Tuza's conjecture. We also prove various bounds on
$\tau^{(m)}(H)/\nu^{(m)}(H)$ for other values of $m$ as well as some bounds on
the fractional analogues of these numbers.
