The critical window for the classical Ramsey-Turan problem

The first application of Szemer,di's powerful regularity method was the following celebrated Ramsey-Turan result proved by Szemer,di in 1972: any K (4)-free graph on n vertices with independence number o(n) has at most edges. Four years later, Bollobas and ErdAs gave a surprising geometric construction, utilizing the isoperimetric inequality for the high dimensional sphere, of a K (4)-free graph on n vertices with independence number o(n) and edges. Starting with Bollobas and ErdAs in 1976, several problems have been asked on estimating the minimum possible independence number in the critical window, when the number of edges is about n (2)/8. These problems have received considerable attention and remained one of the main open problems in this area. In this paper, we give nearly best-possible bounds, solving the various open problems concerning this critical window.