An Erd\H{o}s-Gallai type theorem for uniform hypergraphs

TL;DR

This paper extends the Erd ext{o}s-Gallai theorem to hypergraphs by determining the maximum number of hyperedges in hypergraphs without long Berge paths, completing the characterization for all cases.

Contribution

It proves the remaining case of the hypergraph Erd ext{o}s-Gallai type theorem, establishing a bound on hyperedges for hypergraphs without Berge paths of length r+1.

Findings

Established the maximum number of hyperedges in hypergraphs without Berge paths of length r+1.

Completed the extension of Erd ext{o}s-Gallai theorem to all uniform hypergraph cases.

Proved that exceeding n hyperedges guarantees a Berge path of length r+1.

Abstract

A well-known theorem of Erd\H{o}s and Gallai asserts that a graph with no path of length $k$ contains at most $\frac{1}{2}(k-1)n$ edges. Recently Gy\H{o}ri, Katona and Lemons gave an extension of this result to hypergraphs by determining the maximum number of hyperedges in an $r$-uniform hypergraph containing no Berge path of length $k$ for all values of $r$ and $k$ except for $k=r+1$. We settle the remaining case by proving that an $r$-uniform hypergraph with more than $n$ hyperedges must contain a Berge path of length $r+1$.