Linear paths and trees in uniform hypergraphs

A linear path P ℓ ( k ) is a family of k-sets { F 1 , … , F ℓ } such that | F i ∩ F i + 1 | = 1 and there are no other intersections. We can represent the hyperedges by intervals. With an intensive use of the delta-system method we prove that for t > 0 , k > 3 and sufficiently large n, ( n > n 0 ( k , t ) ), if F is an n-vertex k-uniform family with > ( n − 1 k − 1 ) + ( n − 2 k − 1 ) + ⋯ + ( n − t k − 1 ) , then it contains a linear path of length 2 t + 1 . The only extremal family consists of all edges meeting a given t-set. We also determine ex k ( n , P 2 t ( k ) ) exactly, and the Turán number of any linear tree asymptotically.