Stability of extremal connected hypergraphs avoiding Berge-paths

Abstract

A Berge-path of length $k$ in a hypergraph $\mathcal{H}$ is a sequence $v_1,e_1,v_2,e_2,\dots ,v_k,e_k,v_{k+1}$ of distinct vertices and hyperedges with $v_i,v_{i+1}\in e_i$, for $i\le k$. Füredi, Kostochka and Luo, and independently Győri, Salia and Zamora determined the maximum number of hyperedges in an $n$-vertex, connected, $r$-uniform hypergraph that does not contain a Berge-path of length $k$ provided $k$ is large enough compared to $r$. They also determined the unique extremal hypergraph ${\mathcal{H}}_1$. We prove a stability version of this result by presenting another construction ${\mathcal{H}}_2$ and showing that any $n$-vertex, connected, $r$-uniform hypergraph without a Berge-path of length $k$, that contains more than $|{\mathcal{H}}_2|$ hyperedges must be a sub-hypergraph of the extremal hypergraph ${\mathcal{H}}_1$, provided $k$ is large enough compared to $r$.