The Turán number of a graph H, ex(n, H), is the maximum number of edges in any graph on n vertices which does not contain H as a subgraph. Let P l denote a path on l vertices, and k · P l denote k vertex-disjoint copies of P l . We determine ex(n, k · P 3 ) for n appropriately large, answering in the positive a conjecture of Gorgol. Further, we determine ex (n, k · P l ) for arbitrary l, and n appropriately large relative to k and l. We provide some background on the famous Erdős-Sós conjecture, and conditional on its truth we determine ex(n, H) when H is an equibipartite forest, for appropriately large n.

In 1987, Kolaitis, Prömel and Rothschild proved that, for every fixed r ∈ N, almost every n-vertex K r+1 -free graph is r-partite. In this paper we extend this result to all functions r = r(n) with r (log n) 1/4 . The proof combines a new (close to sharp) supersaturation version of the Erdős-Simonovits stability theorem, the hypergraph container method, and a counting technique developed by Balogh, Bollobás and Simonovits.

Abstract. The Turán number of an r-uniform hypergraph H is the maximum number of edges in any r-graph on n vertices which does not contain H as a subgraph. Let P (r) denote the family of r-uniform loose paths on edges, F (k, l) denote the family of hypergraphs consisting of k disjoint paths from P (r) , and L (r) denote an r-uniform linear path on edges. We determine precisely exr(n; F (k, l)) and exr(n; k · L (r) ), as well as the Turán numbers for forests of paths of differing lengths (whether these paths are loose or linear) when n is appropriately large dependent on k, l, r for r ≥ 3. Our results build on recent results of Füredi, Jiang, and Seiver, who determined the extremal numbers for individual paths, and provide more hypergraphs whose Turán numbers are exactly determined. 1. Introduction and background. Extremal graph theory is that area of combinatorics which is concerned with finding the largest, smallest, or otherwise optimal structures with a given property. Often, the area is concerned with finding the largest (hyper)graph avoiding some subgraph. We build on earlier work of Füredi, Jiang, and Seiver [18], who determined the extremal numbers when the forbidden hypergraph is a single linear path or a single loose path. In this paper, we determine precisely the exact Turán numbers when the forbidden hypergraph is a forest of loose paths or a forest of linear paths; our main results appear in section 2. This is one of only a few papers which gives exact Turán numbers for an infinite family of hypergraphs, in this case, several such families.The Turán number, or extremal number, of an r-uniform hypergraph F is the maximum number of edges in any r-graph H on n vertices which does not contain F as a subgraph. This is a natural generalization of the classical Turán number for 2-graphs; we restrict ourselves to the case of r-uniform hypergraphs, as allowing the extremal number to count edges of different sizes obscures the true extremal structure.Throughout, we use standard terminology and notation (see, e.g., [7]). A hypergraph is a pair H = (V, E) consisting of a set V of vertices and a set E ⊆ P(V ) of edges. If E ⊆