We show that any k-uniform hypergraph with n edges contains two isomorphic edge disjoint subgraphs of size for k = 4, 5 and 6. This is best possible up to a logarithmic factor due to an upper bound construction of Erdős, Pach, and Pyber who show there exist k-uniform hypergraphs with n edges and with no two edge disjoint isomorphic subgraphs with size larger than . Furthermore, our result extends results Erdős, Pach and Pyber who also established the lower bound for k = 2 (eg. for graphs), and of Gould and Rodl who established the result for k = 3. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 2016