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 $v_{i+1}\in e_i,e_{i+1}$ for all $i\in [k]$. 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 subhypergraph of the extremal hypergraph $\mathcal H_1$, provided k is large enough compared to r.