| Summary: | Ramsey-Turán result proved by Szemerédi in 1972: any K [subscript 4-]free graph on n vertices with independence number o(n) has at most (1[over]8+o(1))n[superscript 2] edges. Four years later, Bollobás and Erdős 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 (1[over]8−o(1))n[superscript 2] edges. Starting with Bollobás and Erdős 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[superscript 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.
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