Some new Turan type theorems

Mantel proved in 1907 that if a graph $G$ with $n$ vertices contains no triangle then the number of its edges is at most $\lfloor\frac{n^2}4 \rfloor$. The optimal graph is a complete bipartite graph with equal parts. In 1941 Turan found a generalization: if $G$ contains no copy of the complete graph $K_{k+1}$ then the number of its edges is not more than the complete $k$-partite graph with $k$ equal parts has. In general a Turan type problem asks for the maximum number of edges in a graph $G$ with $n$ vertices containing no copy of a “small” given graph $H$. This maximum is denoted by $\mathrm{ex}(n, H)$. Mantel’s and Turan’s theorems determined $\mathrm{ex}(n, K_3)$ and $\mathrm{ex}(n, K_{k+1})$, respectively. Rademacher observed that if $G$ has $\mathrm{ex}(n, K_3) + 1$ edges then there are suddenly at least $\lfloor\frac n2\rfloor$ triangles, not only one. We prove a strengthening of this statement, namely if the number of edges is $\mathrm{ex}(n,K_3)+1$ and no vertex pins down all triangles then their number is at least $n-2$.
Let $P_k$ denote a path of $k$ vertices. Erd˝os and Gallai determined $\mathrm{ex}(n, P_k)$. Our new result gives the solution of the combined problem: we found $\mathrm{ex}(n,\{K_m, P_k\})$ for large $n$.
The results are jointly achieved with Chuanqi Xiao.