Turán numbers of ordered tight hyperpaths

An ordered hypergraph is a hypergraph $G$ whose vertex set $V(G)$ is linearly ordered. We find the Tur\'an numbers for the $r$-uniform $s$-vertex tight path $P^{(r)}_s$ (with vertices in the natural order) exactly when $r\le s<2r$ and $n$ is even; our results imply $\mathrm{ex}_{>}(n,P^{(r)}_s)=(1-\frac{1}{2^{s-r}} + o(1))\binom{n}{r}$ when $r\le s<2r$. When $r\ge 2s$, the asymptotics of $\mathrm{ex}_{>}(n,P^{(r)}_s)$ remain open. For $r=3$, we give a construction of an $r$-uniform $n$-vertex hypergraph not containing $P^{(r)}_s$ which we conjecture to be asymptotically extremal.