*Abstract.*
A well-known conjecture, often attributed to Ryser, states
that every $r$-partite $r$-uniform hypergraph has cover
number at most $(r - 1)$ times its matching number.
Despite considerable effort, this conjecture remains wide
open, motivating the pursuit of variants of the original
conjecture. Recently, Király and Tóthmérész, and
independently Bustamante and Stein, considered the problem
under the assumption that the hypergraph is
$t$-intersecting, conjecturing that the cover number
$\tau(\mathcal{H})$ of such a hypergraph $\mathcal{H}$ is
at most $r - t$. This conjecture was proven for all $r
\leq 4t-1$.

In this talk, we discuss extensions of this result. For $r
\leq 3t-1$, we prove a tight upper bound on the cover
number of these hypergraphs, showing that they in fact
satisfy $\tau(\mathcal{H}) \leq \lfloor(r - t)/2\rfloor +
1$ and there is a matching construction. We also discuss
upper bounds and constructions for other values of $(r,t)$.
In particular, we extend the range of $t$ for which the
conjectured upper bound of $r-t$ is known to be true,
showing that it holds for all $r < \frac{36}{7}t-5$.

This represents joint work with Anurag Bishnoi, Shagnik
Das and Tibor Szabó.