An Erdős-Gallai type theorem for uniform hypergraphs

Abstract A well-known theorem of Erdős and Gallai (1959) [ 1 ] asserts that a graph with no path of length k contains at most 1 2 ( 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 .