On the minimal degree condition of graphs implying some properties of subgraphs

ErdH{o}s posed the problem of finding conditions on a graph $G$ that imply the largest number of edges in a triangle-free subgraph is equal to the largest number of edges in a bipartite subgraph. We generalize this problem to general cases. Let $delta_r$ be the least number so that any graph $G$ on $n$ vertices with minimum degree $delta_rn$ has the property $P_{r-1}(G)=K_rf(G),$ where $P_{r-1}(G)$ is the largest number of edges in an $(r-1)$-partite subgraph and $K_rf(G)$ is the largest number of edges in a $K_r$-free subgraph. We show that $frac{3r-4}{3r-1}<delta_rlefrac{4(3r-7)(r-1)+1}{4(r-2)(3r-4)}$ when $rge4.$ In particular, $delta_4le 0.9415.$