Let $f(k,r)$ be minimal such that if $A_1,A_2,\ldots$ is a family of sets, all of size $k$, such that for every collection of $r$ of the $A_is$ there is some pair $\{x,y\}$ which intersects all of the $A_j$, then there is some set of size $f(k,r)$ which intersects all of the sets $A_i$. Is it true that

$$f(k,7)=(1+o(1))\frac{3}{4}k?$$

Is it true that for any $r\geq 3$ there exists some constant $c_r$ such that

$$f(k,r)=(1+o(1))c_rk?$$