On the maximum size of connected hypergraphs without a path of given length

TL;DR

This paper asymptotically determines the maximum number of hyperedges in large connected r-uniform hypergraphs without long Berge paths, revealing differences from the graph case due to connectivity constraints.

Contribution

It provides the first asymptotic bounds for hypergraph maximum size under path length restrictions, highlighting the impact of connectivity.

Findings

Maximum hyperedges are asymptotically characterized for large hypergraphs.

Connectivity reduces the maximum possible number of hyperedges.

Results differ from the classical graph case due to hypergraph structure.

Abstract

In this note we asymptotically determine the maximum number of hyperedges possible in an $r$-uniform, connected $n$-vertex hypergraph without a Berge path of length $k$, as $n$ and $k$ tend to infinity. We show that, unlike in the graph case, the multiplicative constant is smaller with the assumption of connectivity.