Turán number of disjoint Berge paths
Yiyan Zhan, Xiamiao Zhao, Mei Lu
TL;DR
The paper determines the exact Turán number ex_r(n, Berge- kP_ℓ) for large n, in the regime k≥2, r≥3, with ℓ' = floor((ℓ+1)/2) and 2ℓ'≥ r+7. The authors build tight lower-bound constructions using a two-part vertex partition to avoid Berge- kP_ℓ and prove matching upper bounds by first solving the k=2 case (with a detailed analysis of longest Berge-paths and Berge-cycles) and then extending via induction on k. A key technical component is demonstrating that large Berge-P_ℓ-structures compel abundant Berge-common-neighbor configurations, enabling a global counting argument that constrains the extremal hypergraph’s vertex set and forces the prescribed extremal form. The result extends Erdős-Gallai type Turán theory to Berge-paths in r-uniform hypergraphs and provides a precise, asymptotically tight formula dependent on k, ℓ, and r, with implications for linear forests and related extremal problems.
Abstract
For a graph $F$, an $r$-uniform hypergraph $\mathcal{H}$ is a $\text{Berge-} F$ if there is a bijection $φ: E(F)\to E(\mathcal{H})$ such that $e\subseteq φ(e)$ for each $e\in E(F)$. When $F$ is a path, we call $\text{Berge-} F$ as Berge path. Let $\mathcal{F}$ be a family of $r$-graphs. An $r$-graph $\mathcal{H}$ is called $\mathcal{F}$-free if $\mathcal{H}$ does not contain any member in $\mathcal{F}$ as a subhypergraph. The Turán number $\mathrm{ex}_{r}(n,\mathcal{F})$ is the maximum number of hyperedges in an $\mathcal{F}$-free $r$-graph on $n$ vertices. The Turán number of Berge paths has received widespread attention. In this paper, we determine the exact value of $\mathrm{ex}_r(n,\text{Berge-} kP_{\ell})$ when $n$ is large enough for $k\geq 2$, $r\ge 3$, $\ell'\geq r$ and $2\ell'\geq r+7$, where $\ell'=\left\lfloor\frac{\ell+1}{2}\right\rfloor$.