  A famous conjecture (usually called Ryser's conjecture) that appeared in the
  Ph.D thesis of his student, J.~R.~Henderson [15], states that for an
$r$-uniform $r$-partite hypergraph $\mathcal{H}$, the inequality
$\tau(\mathcal{H})\le(r-1)\cdot \nu(\mathcal{H})$ always holds.
  This conjecture is widely open, except in the case of $r=2$, when it is
equivalent to K\H onig's theorem [18], and in the case of $r=3$, which was
proved by Aharoni in 2001 [3].
  Here we study some special cases of Ryser's conjecture. First of all the most
studied special case is when $\mathcal{H}$ is intersecting. Even for this
special case, not too much is known: this conjecture is proved only for $r\le
5$ in [10,21]. For $r>5$ it is also widely open.
  Generalizing the conjecture for intersecting hypergraphs, we conjecture the
following. If an $r$-uniform $r$-partite hypergraph $\mathcal{H}$ is
$t$-intersecting (i.e., every two hyperedges meet in at least $t<r$ vertices),
then $\tau(\mathcal{H})\le r-t$. We prove this conjecture for the case $t>
r/4$.
  Gy\'arf\'as [10] showed that Ryser's conjecture for intersecting hypergraphs
is equivalent to saying that the vertices of an $r$-edge-colored complete graph
can be covered by $r-1$ monochromatic components.
  Motivated by this formulation, we examine what fraction of the vertices can
be covered by $r-1$ monochromatic components of \emph{different} colors in an
$r$-edge-colored complete graph. We prove a sharp bound for this problem.
  Finally we prove Ryser's conjecture for the very special case when the
maximum degree of the hypergraph is two.
