Turán problems for star-path forests in hypergraphs

Abstract: 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 Tur\'{a}n number of $\mathcal{F}$ is the maximum number of edges in any $\mathcal{F}$-free $r$-graph on $n$ vertices, and the linear Tur\'{a}n number of $\mathcal{F}$ is defined as the Tur\'{a}n number of $\mathcal{F}$ in linear host hypergraphs. An $r$-uniform linear path $P^r_\ell$ of length $\ell$ is an $r$-graph with edges $e_1,\dots,e_\ell$ such that $|V(e_i)\cap V(e_j)|=1$ if $|i-j|=1$, and $V(e_i)\cap V(e_j)=\emptyset$ for $i\neq j$ otherwise. Gy\'{a}rf\'{a}s et al. [\textit{European J. Combin.} (2022) 103435] obtained an upper bound for the linear Tur\'{a}n number of $P_\ell^3$. In this paper, an upper bound for the linear Tur\'{a}n number of $P_\ell^r$ is obtained, which generalizes the known result of $P_\ell^3$ to any $P_\ell^r$. Furthermore, some results for the linear Tur\'{a}n number and Tur\'{a}n number of several linear star-path forests are obtained.