Extremal results on Berge disjoint paths

Abstract

The well-known Erdős-Gallai Theorem gave the Turán number of paths. Bushaw and Kettle generalized this result to consider the Turán number of disjoint paths. Since then, many studies are focused on the Turán number of linear forest. For a graph $F$, an $r$-uniform hypergraph $\mathcal{H}$ is a $\text{Berge-}F$ if there is a bijection $\ddot{I}•:E(F)\to E(\mathcal{H})$ such that $e\subseteq \ddot{I}•(e)$ for each $e\in E(F)$. When $F$ is a path, we call $\text{Berge-}F$ a Berge path. The Turán number of Berge paths was initially studied by Győri, Katona and Lemons. They gave the value of ${\text{ex}}_r(n,\text{Berge-}{P}_{\ell })$ for $\ell >r+1$. This result is a generalization of Erdős-Galli Theorem. Since then, the Turán number of Berge paths has received widespread attention. Recently, Zhou, Gerbner and Yuan initially studied the Turán number of Berge disjoint paths and for the cases when all the paths have odd length. In this paper, we give a more general result, which gives the exact value of ${\mathrm{ex}}_r(n,\text{Berge-}k{P}_{\ell })$ for all $k\ge 2$, $r\ge 3$, and $\ell \ge r+7$.