  Let $P^k_\ell$ denote the loose $k$-path of length $\ell$ and let define
$f^k_\ell(n,m)$ as the minimum value of $\Delta(H)$ over all $P^k_\ell$-free
$k$-graphs $H$ with $n$ vertices and $m$ edges. In the paper we study the
behavior of $f^4_2(n,m)$ and $f^3_3(n,m)$ and characterize the structure of
extremal hypergraphs. In particular, it is shown that when $m\sim n^2/8$ the
value of each of these functions drops down from $\Theta(n^2)$ to $\Theta(n)$.
