Linear three-uniform hypergraphs with no Berge path of given length

TL;DR

This paper extends the Erd ext{"o}s-Gallai Theorem to linear 3-uniform hypergraphs, establishing an upper bound on the number of hyperedges without long Berge paths.

Contribution

It proves a new upper bound for the number of hyperedges in linear 3-uniform hypergraphs avoiding Berge paths of a given length.

Findings

Upper bound of
76(n) for hyperedges without Berge paths of length k

Extension of Erd ext{"o}s-Gallai Theorem to hypergraphs

Results applicable for k  4

Abstract

Extensions of Erd\H{o}s-Gallai Theorem for general hypergraphs are well studied. In this work, we prove the extension of Erd\H{o}s-Gallai Theorem for linear hypergraphs. In particular, we show that the number of hyperedges in an $n$-vertex $3$-uniform linear hypergraph, without a Berge path of length $k$ as a subgraph is at most $\frac{(k-1)}{6}n$ for $k\geq 4$.