The Turán number of the square of a path

Abstract The Turan number of a graph H , ex ( n , H ) , is the maximum number of edges in a graph on n vertices which does not have H as a subgraph. Let P k be the path with k vertices, the square P k 2 of P k is obtained by joining the pairs of vertices with distance one or two in P k . The powerful theorem of Erdős, Stone and Simonovits determines the asymptotic behavior of ex ( n , P k 2 ) . In the present paper, we determine the exact value of ex ( n , P 5 2 ) and ex ( n , P 6 2 ) and pose a conjecture for the exact value of ex ( n , P k 2 ) .