Tight instance for unweighted maximum coverage problem?
Here is one example:
take the universe   $$U=\{u_1,\dots,u_n,v_1,\dots,v_n\}$$
$S_i=\{u_i,v_i\}$ for $i=1,\dots,n$ , and  $S_{n+1}=\{u_1,\dots,u_n\}$, $m=n+1$, $k=n$