Given a family $\mathcal{F}$ of sets over ground set $X$, let $\tau(\mathcal{F})$ be the **transversal number** (aka blocking number), that is the cardinality of the smallest set of points $E \subseteq X$ such that every set in $\mathcal{F}$ meets $E$.

Lovasz (Problem 13.25 in his Problems & Exercises books) has proved that if $F$ is an $r$-uniform family of sets that every $k$ sets in $F$ intersect, then: $$ \tau(\mathcal{F}) \leq \frac{r-1}{k-1}+1. $$

I wonder whether it is possible to strengthen the conclusion if we assume that $\mathcal{F}$ is $k$-wise $t$-intersecting (that is, every $k$ sets in $\mathcal{F}$ intersect in at least $t$ points).

I can obtain the following by mimicking the proof, but I feel there ought to be a better result somewhere: $$ \tau(\mathcal{F}) \leq \frac{r-t}{k-1}+1. $$