Maximizing the number of cliques in $K_{r+1}$-free graphs with forbidden properties

Ferrero and Lesniak in 2018 found the maximum numbers of edges in $r$-partite non-Hamiltonian graphs. Recently we found the maximum numbers of edges and $t$-cliques in $K_{r+1}$-free graphs (1) that are not Hamiltonian or (2) that satisfy a condition on low-degree vertices related to Pósa’s theorem. Applying theorem (2), here we extend theorem (1) from Hamiltonicity to other properties. We determine the maximum numbers of edges and $t$-cliques in $K_{r+1}$-free graphs that avoid one of the following properties: traceability, Hamiltonian-connectedness, $k$-path Hamiltonicity, $k$-Hamiltonicity, $k$-Hamiltonian-connectedness, and $k$-connectedness. We find all extremal graphs having the maximum numbers of edges. On the way, we prove upper bounds on the numbers of edges and $t$-cliques in $K_{r+1}$-free graphs that avoid an arbitrary stable property that holds for sufficiently large complete graphs.