Hypergraph Extensions of the Erdos-Gallai Theorem

Our goal is to extend the following result of Erd˝ os and Gallai for hypergraphs: Theorem 1 (Erd˝ os-Gallai [1]) Let G be a graph on n vertices containing no path of length k. Then e(G) ≤ 1 2 (k − 1)n. Equality holds iff G is the disjoint union of complete graphs on k vertices. We consider several generalizations of this theorem for hypergraphs. This is due to the fact that there are several possible ways to define paths in hypergraphs. One such definition of paths in hypergraphs is due to Berge. Definition 2 A Berge path of length k in a hypergraph is a collection of k hy-We find the extremal sizes of r-uniform hypergraphs avoiding Berge cycles of length k. Interestingly, the size of the extremal hypergraphs depend on the relationship between r and k. Specifically, we distinguish between the cases when k ≤ r and when k > r.