The Turán number of Berge paths in connected hypergraphs

Source abstract

A Berge path in a hypergraph is an alternating sequence of distinct vertices and edges $v_1,e_1, v_2, \ldots, e_{k-1}, v_{k}$ such that $v_i, v_{i+1} \in e_i$ for all $i \in [k-1]$. Győri, Katona, and Lemons determined the maximum number edges in $r$-uniform, $n$-vertex hypergraphs without Berge paths on $k$ vertices. The extremal examples for their bounds are disconnected hypergraphs in which each component has either too few edges or too few vertices to contain such a Berge path. We consider the setting in which the host hypergraph is connected. Let $\operatorname{ex}^{\mathrm{conn}}_r(n,\mathcal{BP}_k)$ denote the maximum number of edges in an $n$-vertex, $r$-uniform, connected hypergraph that does not contain a Berge path on $k$ vertices. This number was determined by Zhang, Broersma, Győri, Tompkins, Wang for $k \leq r+2$, and Zhao, Gerbner, Zhou for $k \geq 2r+2$. In this paper, we resolved the remaining cases $r+3 \leq k \leq 2r+1$ when $n$ is sufficiently large, confirming a conjecture of Zhang, et al.