Extremal graphs without long paths and large cliques

Abstract

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 $ex\left(n,F\right)$. The same maximum under the additional condition that the graphs are connected is ${ex}_{\mathrm{conn}}\left(n,F\right)$. Let $P_k$ be the path on $k$ vertices, $K_m$ be the clique on $m$ vertices. We determine $ex(n,\{P_k,K_m\})$ if $k>2m-1$ and ${ex}_{\mathrm{conn}}(n,\{P_k,K_m\})$ if $k>m$ for sufficiently large $n$.