Turan Numbers Of Multiple Paths And Equibipartite Forests

The Turan 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 let k center dot P(l) denote k vertex-disjoint copies of P(l). We determine ex(n, k center dot P(3)) for n appropriately large, answering in the positive a conjecture of Gorgol. Further, we determine ex(n, k center dot P(l)) for arbitrary l, and n appropriately large relative to k and l. We provide some background on the famous Erdos-Sos conjecture, and conditional on its truth we determine ex(n, H) when H is an equibipartite forest, for appropriately large n.