  A famous conjecture of Tuza \cite{tuza} is that the minimal number of edges
needed to cover all triangles in a graph is at most twice the maximal number of
edge-disjoint triangles. We propose a wider setting for this conjecture.
  For a hypergraph $H$ let $\nu^{(m)}(H)$ be the maximal size of a collection
of edges, no two of which share $m$ or more vertices, and let $\tau^{(m)}(H)$
be the minimal size of a collection $C$ of sets of $m$ vertices, such that
every edge in $H$ contains a set from $C$. We conjecture that the maximal ratio
$\tau^{(m)}(H)/\nu^{(m)}(H)$ is attained in hypergraphs for which
$\nu^{(m)}(H)=1$. This would imply, in particular, the following generalization
of Tuza's conjecture: if $H$ is $3$-uniform, then $\tau^{(2)}(H)/\nu^{(2)}(H)
\le 2$. (Tuza's conjecture is the case in which $H$ is the set of all triples
of vertices of triangles in the graph). We show that most known results on
Tuza's conjecture go over to this more general setting. We also prove some
general results on the ratio $\tau^{(m)}(H)/\nu^{(m)}(H)$, and study the
fractional versions and the case of $k$-partite hypergraphs.
