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The Tur\'an number of a graph H, ex(n;H), is the maximum number of edges inany graph on n vertices which does not contain H as a subgraph. Let P_l denotea path on l vertices, and kP_l denote k vertex-disjoint copies of P_l. Wedetermine ex(n, kP_3) for n appropriately large, answering in the positive aconjecture of Gorgol. Further, we determine ex (n, kP_l) for arbitrary l, and nappropriately large relative to k and l. We provide some background on thefamous Erd\H{o}s-S\'os conjecture, and conditional on its truth we determineex(n;H) when H is an equibipartite forest, for appropriately large n.

Tur\`an numbers of Multiple Paths and Equibipartite Trees