We extend the Erdős-Gallai Theorem for Berge paths in r-uniform hypergraphs. We also find the extremal hypergraphs avoiding t-tight paths of a given length and consider this extremal problem for other definitions of paths in hypergraphs.

Recently, the authors gave upper bounds for the size of 3-uniform hypergraphs avoiding a given odd cycle using the definition of a cycle due to Berge. In the present paper we extend this bound to m-uniform hypergraphs (for all m ≥ 3), as well as m-uniform hypergraphs avoiding a cycle of length 2k. Finally we consider non-uniform hypergraphs avoiding cycles of length 2k or 2k + 1. In both cases we can bound |h| by O(n1+1/k) under the assumption that all h ∈ ε() satisfy |h| ≥ 4k2.

Given two graphs H and G, let H(G) denote the number of subgraphs of G isomorphic to H. We prove that if H is a bipartite graph with a one-factor, then for every triangle-free graph G with n vertices H(G) < H(T2(n)), where T2(n) denotes the complete bipartite graph of n vertices whose eolour classes are as equal as possible. We also prove that if K is a complete t-partite graph of m vertices, r > t, n > max(m, r -1), then there exists a complete (r -1)-partite graph G* with n vertices such that K(G) < K(G*) holds for every Kr-free graph G with n vertices. In particular, in the class of all/(,-free graphs with n vertices the complete balanced (r -1)-partite graph T,_l(n ) has the largest number of subgraphs isomorphic to K t (t < r), C,, K2.a. These generalize some theorems of Turhn, Erdfs and Sauer.' a+

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