OPEN

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
\[f(k,r)=(1+o(1))c_rk?\]

A problem of Erdős, Fon-Der-Flaass, Kostochka, and Tuza [EFKT92], who proved that $f(k,3)=2k$ and $f(k,4)=\lfloor 3k/2\rfloor$ and $f(k,5)=\lfloor 5k/4\rfloor$, and further that $f(k,6)=k$.