Turán 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 Pl denote a path on l vertices, and let k ⋠Pl denote k vertex-disjoint copies of Pl. We determine ex(n, k ⋠P3) for n appropriately large, answering in the positive a conjecture of Gorgol. Further, we determine ex(n, k ⋠Pl) for arbitrary l, and n appropriately large relative to k and l. We provide some background on the famous ErdA‘s-Sos conjecture, and conditional on its truth we determine ex(n, H) when H is an equibipartite forest, for appropriately large n.