An Erdős-Gallai type theorem for uniform hypergraphs

A well-known theorem of Erdős and Gallai (1959) [1] asserts that a graph with no path of length k contains at most 12(k−1)n edges. Recently Győri et al. (2016) 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 edges must contain a Berge path of length r+1.