Isomorphic Subgraphs in a Graph

At the combinatorics meeting of this Proceedings J . Schönheim posed the following problem : Is it true, that every graph of n edges has two (not necessarily induced isomorphic edge disjoint subgraphs with say f edges? In the present note we answer this question in the affirmative . In fact we prove that every graph of n edges contains two isomorphic edge disjoint subgraphs with cn 2/3 edges and apart from the constant factor this result is best possible . Various generalizations are considered . For any hypergraph X, let V(X) and E(X) denote the set of vertices and the set of (hyper)edges of X, respectively. JE(X)l will be called the size of X . Given any natural numbers r,s > 2, let fr, ;,(n) denote the maximum integer f such that in every r-uniform hypergraph X of size n one can find s pairwise edge-disjoint isomorphic subhypergraphs Xl, X2 , . . ., X, C X of size f . We can summarize our results in the following .