Turán problems for star-path forests in hypergraphs

An $r$-uniform hypergraph ($r$-graph for short) is linear if any two edges intersect at most one vertex. Let $\mathcal{F}$ be a given family of $r$-graphs. An $r$-graph $H$ is called $\mathcal{F}$-free if $H$ does not contain any member of $\mathcal{F}$ as a subgraph. The Turán number of $\mathcal{F}$ is the maximum number of edges in any $\mathcal{F}$-free $r$-graph on $n$ vertices, and the linear Turán number of $\mathcal{F}$ is defined as the Turán number of $\mathcal{F}$ in linear host hypergraphs. An $r$-uniform linear path $P_{\mathrm{\ell }}^r$ of length $\mathrm{\ell }$ is an $r$-graph with edges $e_1,\dots ,e_{\mathrm{\ell }}$ such that $\mathrm{\mid }V(e_i)\cap V(e_j)\mathrm{\mid }=1$ if $\mathrm{\mid }i-j\mathrm{\mid }=1$, and $V(e_i)\cap V(e_j)=\mathrm{\varnothing }$ for $i\mathrm{\ne }j$ otherwise. Gyárfás et al. [$\mathit{\text{European}}\mathit{\text{J.}}\mathit{\text{Combin.}}$ (2022) 103435] obtained an upper bound for the linear Turán number of $P_{\mathrm{\ell }}^3$. In this paper, an upper bound for the linear Turán number of $P_{\mathrm{\ell }}^r$ is obtained, which generalizes the known result of $P_{\mathrm{\ell }}^3$ to any $P_{\mathrm{\ell }}^r$. Furthermore, some results for the linear Turán number and Turán number of several linear star-path forests are obtained.