Connected Hypergraphs without long Berge paths

TL;DR

This paper characterizes the structure of connected hypergraphs that maximize hyperedges without containing long Berge paths, extending previous results to broader conditions.

Contribution

It generalizes a known extremal hypergraph result to connected hypergraphs with specific parameters, identifying the unique extremal structure.

Findings

Determines the extremal structure for hypergraphs without long Berge paths.

Establishes the maximum number of hyperedges in such hypergraphs.

Provides conditions under which the extremal structure is unique.

Abstract

We generalize a result of Balister, Gy{\H{o}}ri, Lehel and Schelp for hypergraphs. We determine the unique extremal structure of an $n$-vertex, $r$-uniform, connected, hypergraph with the maximum number of hyperedges, without a $k$-Berge-path, where $n \geq N_{k,r}$, $k\geq 2r+13>17$.