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Turàn numbers of Multiple Paths and Equibipartite Trees

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 kP_l denote k vertex-disjoint copies of P_l. We determine ex(n, kP_3) for n appropriately large, answering in the positive a conjecture of Gorgol. Further, we determine ex (n, kP_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.