Exact solution of the hypergraph Turán problem for k-uniform linear paths

A k-uniform linear path of length l, denoted by P (k) l , is a family of k-sets {F1, . . . , Fl} such that |Fi ∩ Fi+1| = 1 for each i and Fi ∩ Fj = ∅ 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 exk(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 exk(n, P (k) l ) exactly for all fixed l ≥ 1, k ≥ 4, and sufficiently large n. We show that exk(n,P (k) 2t+1) = ( n− 1 k − 1 ) + ( n− 2 k − 1 ) + . . .+ ( n− t k − 1 ) . The only extremal family consists of all the k-sets in [n] that meet some fixed set of t vertices. We also show that ex(n,P (k) 2t+2) = ( n− 1 k − 1 ) + ( n− 2 k − 1 ) + . . .+ ( n− t k − 1 ) + ( n− t− 2 k − 2 ) , and describe the unique extremal family. Stability results on these bounds and some related results are also established.