Tur\'{a}n problems for star-path forests in hypergraphs

An r-uniform hypergraph (r-graph) is linear if any two edges intersect at most one vertex. Let F be a given family of r-graphs. A hypergraph H is called F-free if H does not contain any hypergraphs in F. The Tur\'{a}n number exᵣ(n,F) of F is defined as the maximum number of edges of all F-free r-graphs on n vertices, and the linear Tur\'{a}n number exˡⁱⁿᵣ(n,F) of F is defined as the Tur\'{a}n number of F in linear host hypergraphs. An r-uniform linear path Pʳ_ of length is an r-graph with edges e₁,⋯,e_ such that |V(eᵢ)∩ V(eⱼ)|=1 if $|i-j|=1$, and V(eᵢ)∩ V(eⱼ)=∅ for i≠ j otherwise. Gy\'{a}rf\'{a}s et al. [Linear Tur\'{a}n numbers of acyclic triple systems, European J. Combin., 2022, 103435] obtained an upper bound for the linear Tur\'{a}n number of P_³. In this paper, an upper bound for the linear Tur\'{a}n number of P_ʳ is obtained, which generalizes the result of P_³ to any P_ʳ. Furthermore, some results for the linear Tur\'{a}n number and Tur\'{a}n number of some linear star-path forests are obtained.