Paths in hypergraphs: a rescaling phenomenon

Let $P_{\mathrm{\ell }}^k$ denote the loose $k$-path of length $\mathrm{\ell }$ and let define $f_{\mathrm{\ell }}^k(n,m)$ as the minimum value of $\mathrm{\Delta }(H)$ over all $P_{\mathrm{\ell }}^k$-free $k$-graphs $H$ with $n$ vertices and $m$ edges. In the paper we study the behavior of $f_2^4(n,m)$ and $f_3^3(n,m)$ and characterize the structure of extremal hypergraphs. In particular, it is shown that when $m\sim n^2\mathrm{/}8$ the value of each of these functions drops down from $\mathrm{\Theta }(n^2)$ to $\mathrm{\Theta }(n)$.