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

Erdős 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 δr be the least number so that any graph G on n vertices with minimum degree δrn has the property Pr−1(G) = Krf(G), where Pr−1(G) is the largest number of edges in an (r − 1)-partite subgraph and Krf(G) is the largest number of edges in a Kr-free subgraph. We show that 3r−4 3r−1 < δr ≤ 4(3r−7)(r−1)+1 4(r−2)(3r−4) when r ≥ 4. In particular, δ4 ≤ 0.9415.