On the minimal degree implying equality of the largest triangle-free and bipartite subgraphs

Erdős posed the problem of finding conditions on a graph G that imply t(G)=b(G), where t(G) is the largest number of edges in a triangle-free subgraph and b(G) is the largest number of edges in a bipartite subgraph. Let δc be the least number so that any graph G on n vertices with minimum degree δcn has t(G)=b(G). Extending results of Bondy, Shen, Thomassé and Thomassen we show that 0.75⩽δc<0.791.