A k-uniform linear path of length ?, denoted by P_?~((k)), is a family of k-sets {F_1,...,F_?} such that |F_i ∩ F_(i+1)|=1 for each i and F_i ∩ F_j =? whenever |i-j|>1. Given a k-uniform hypergraph H and a positive integer n, the k-uniform hypergraph Turán number of H, denoted by ex_k(n,H), is the maximum number of edges in a k-uniform hypergraph F on n vertices that does not contain H as a subhypergraph. With an intensive use of the delta-system method, we determine ex_k(n,P_?~((k))) exactly for all fixed ?≥1,k≥4, and sufficiently large n. We show that ex_k(n, P_(2t+1)~((k))=(~(n - 1)_(k - 1)+~(n - 2)_(k - 1)+...+~(n - t)_(k - 1).
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