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The critical window for the classical Ramsey-Turán problem

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Abstract

The first application of Szemerédi’s powerful regularity method was the following celebrated Ramsey-Turán result proved by Szemerédi in 1972: any K 4-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 K 4-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 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.

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Authors and Affiliations

  1. Department of Mathematics, MIT, Cambridge, MA, 02139-4307, USA

    Jacob Fox & Yufei Zhao

  2. Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA, 15213, USA

    Po-Shen Loh

Authors
  1. Jacob Fox
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  2. Po-Shen Loh
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  3. Yufei Zhao
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Correspondence to Po-Shen Loh.

Additional information

Research supported by a Simons Fellowship, NSF grant DMS-1069197, and the MIT NEC Research Corporation Fund.

Research supported by NSF grant DMS-1201380, an NSA Young Investigators Grant and a USA-Israel BSF Grant.

Research supported by an Akamai Presidential Fellowship.

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Fox, J., Loh, PS. & Zhao, Y. The critical window for the classical Ramsey-Turán problem. Combinatorica 35, 435–476 (2015). https://doi.org/10.1007/s00493-014-3025-3

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