A well-known conjecture, often attributed to Ryser, states that the cover number of an r-partite r-uniform hypergraph is at most $r-1$ times larger than its matching number. Despite considerable effort, particularly in the intersecting case, this conjecture remains wide open, motivating the pursuit of variants of the original conjecture. Recently, Bustamante and Stein and, independently, Király and Tóthmérész 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 $r-t$. In these papers, it was proven that the conjecture is true for $r\le 4t-1$, but also that it need not be sharp; when $r=5$ and $t=2$, one has $\tau (\mathcal{H})\le 2$.

We extend these results in two directions. First, for all $t\ge 2$ and $r\le 3t-1$, we prove a tight upper bound on the cover number of these hypergraphs, showing that they in fact satisfy $\tau (\mathcal{H})\le \lfloor (r-t)/2\rfloor +1$. Second, we extend the range of t for which the conjecture is known to be true, showing that it holds for all $r\le \frac{36}{7}t-5$. We also introduce several related variations on this theme. As a consequence of our tight bounds, we resolve the problem for k-wise t-intersecting hypergraphs, for all $k\ge 3$ and $t\ge 1$. We further give bounds on the cover numbers of strictly t-intersecting hypergraphs and the s-cover numbers of t-intersecting hypergraphs.

通常归因于Ryser的一个著名猜想指出，r-部分r-一致超图的覆盖数最多为$\mathrm{[R}-\mathrm{1个}$比其匹配数大一倍。尽管付出了巨大的努力，特别是在相交的情况下，这个猜想仍然是敞开的，从而激发了对原始猜想的各种追求。最近，Bustamante和Stein以及独立的Király和Tóthmérész在假设超图是t相交的假设下考虑了这个问题，推测覆盖数是$\tau （\mathcal{H}）$ 这样的超图 $\mathcal{H}$ 最多 $\mathrm{[R}-Ť$。在这些论文中，证明了对于$\mathrm{[R}\le 4Ť-\mathrm{1个}$，但也不必太尖锐；什么时候$\mathrm{[R}=5$ 和 $Ť=2$，有 $\tau （\mathcal{H}）\le 2$。

我们将这些结果扩展到两个方向。首先，对于所有人$Ť\ge 2$ 和 $\mathrm{[R}\le 3Ť-\mathrm{1个}$，我们证明了这些超图的覆盖数有一个严格的上限，表明它们实际上满足了 $\tau （\mathcal{H}）\le \lfloor （\mathrm{[R}-Ť）/2\rfloor +\mathrm{1个}$。其次，我们扩展了已知猜想为真的t的范围，表明它对所有$\mathrm{[R}\le \frac{36}{7}Ť-5$。我们还介绍了与此主题相关的几种相关变化。由于我们的紧密边界，我们解决了所有k个t相交超图的问题$ķ\ge 3$ 和 $Ť\ge \mathrm{1个}$。我们进一步给出严格的t相交超图的覆盖数和t相交超图的s覆盖数的界限。