, Volume 35, Issue 4, pp 435–476

The critical window for the classical Ramsey-Turán problem

Original Paper


The first application of Szemerédi’s powerful regularity method was the following celebrated Ramsey-Turán result proved by Szemerédi in 1972: any K4-free graph on n vertices with independence number o(n) has at most \((\tfrac{1} {8} + o(1))n^2\) edges. Four years later, Bollobás and Erdős gave a surprising geometric construction, utilizing the isoperimetric inequality for the high dimensional sphere, of a K4-free graph on n vertices with independence number o(n) and \((\tfrac{1} {8} - o(1))n^2\) 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 n2/8. 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.

Mathematics Subject Classication (2000)

05C35 05C55 05D40 


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Copyright information

© János Bolyai Mathematical Society and Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Department of MathematicsMITCambridgeUSA
  2. 2.Department of Mathematical SciencesCarnegie Mellon UniversityPittsburghUSA

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