Connected Turán numbers for Berge paths in hypergraphs

Let $\mathcal{F}$ be a family of $r$-uniform hypergraphs. Denote by ${\text{ex}}_r^{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{n}}(n,\mathcal{F})$ the maximum number of hyperedges in an $n$-vertex connected $r$-uniform hypergraph which contains no member of $\mathcal{F}$ as a subhypergraph. Denote by $\mathcal{B}{C}_k$ the Berge cycle of length $k$, and by $\mathcal{B}{P}_k$ the Berge path of length $k$. Füredi, Kostochka and Luo, and independently Győri, Salia and Zamora determined ${\text{ex}}_r^{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{n}}(n,\mathcal{B}{P}_k)$ provided $k$ is large enough compared to $r$ and $n$ is sufficiently large. For the case $k\le r$, Kostochka and Luo obtained an upper bound for ${\text{ex}}_r^{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{n}}(n,\mathcal{B}{P}_k)$. In this paper, we continue investigating the case $k\le r$. We precisely determine ${\text{ex}}_r^{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{n}}(n,\mathcal{B}{P}_k)$ when $n$ is sufficiently large and $n$ is not a multiple of~$r$. For the case $k=r+1$, we determine ${\text{ex}}_r^{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{n}}(n,\mathcal{B}{P}_k)$ asymptotically.