Loose paths in random ordered hypergraphs

We consider the length of {\em ordered loose paths} in the random $r$-uniform hypergraph $H=H^{(r)}(n,p)$. A ordered loose path is a sequence of edges $E_1,E_2,\dots ,E_{\mathrm{\ell }}$ where $\max \{j\in E_i\}=\min \{j\in E_{i+1}\}$ for $1\le i<\mathrm{\ell }$. We establish fairly tight bounds on the length of the longest ordered loose path in $H$ that hold with high probability.