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

Abstract

Extensions of Erdős-Gallai Theorem for general hypergraphs are well studied. In this work, we prove the extension of Erdős-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 $k\ge 4$. The bound is sharp for infinitely many k and n.