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 $F$ be a given family of r-graphs. An r-graph H is called $F$-free if H does not contain any member of $F$ as a subgraph. The Turán number of $F$ is the maximum number of edges in any $F$-free r-graph on n vertices, and the linear Turán number of $F$ is defined as the Turán number of $F$ in linear host hypergraphs. An r-uniform linear path $P_{\ell }^r$ of length ℓ is an r-graph with edges $e_1,\dots ,e_{\ell }$ such that $\left|V\right(e_i)\cap V(e_j\left)\right|=1$ if $|i-j|=1$, and $V\left(e_i\right)\cap V\left(e_j\right)=\varnothing$ for $i\ne j$ otherwise. Gyárfás et al. (2022) [9] obtained an upper bound for the linear Turán number of $P_{\ell }^3$. In this paper, an upper bound for the linear Turán 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 Turán number and Turán number of several linear star-path forests are obtained.