3-uniform linear hypergraphs without a long Berge path

Abstract

Extensions of the Erd\H{o}s-Gallai theorem for general hypergraphs are well studied. In this work, we prove the extension of the 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 $k\geq 4$. This is an extended abstract for EUROCOMB23 of the manuscript arXiv:2211.16184.