Extremal graphs without long paths and large cliques

Let F be a family of graphs. A graph is called F-free if it does not contain any member of F as a subgraph. The Turán number of F is the maximum number of edges in an n-vertex F-free graph and is denoted by ex(n,F). The same maximum under the additional condition that the graphs are connected is exconn(n,F). Let Pk be the path on k vertices, Km be the clique on m vertices. We determine ex(n,{Pk,Km}) if k>2m−1 and exconn(n,{Pk,Km}) if k>m for sufficiently large n.