Stability of extremal connected hypergraphs avoiding Berge-paths

TL;DR

This paper establishes a stability result for the maximum size of connected, r-uniform hypergraphs avoiding long Berge-paths, showing near-extremal hypergraphs are structurally close to a known extremal configuration.

Contribution

It proves a stability version of the maximum edge count in hypergraphs avoiding Berge-paths, identifying conditions under which hypergraphs are close to the extremal structure.

Findings

Identified a new hypergraph construction $\\mathcal{H}_2$

Proved hypergraphs exceeding $|\mathcal{H}_2|$ edges are subgraphs of the extremal hypergraph

Established stability conditions for large Berge-path avoidance

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 $i \le k$. F\"uredi, Kostochka and Luo, and independently Gy\H{o}ri, 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 $r$.