The critical window for the classical Ramsey-Turán problem
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Fox_The critical.pdf
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361.35 KB
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Author(s)
Fox, Jacob
Loh, Po-Shen
Zhao, Yufei
Date Issued
October 2014
Journal
Combinatorica
Publisher
Springer-Verlag/Bolyai Society
Citation
Fox, Jacob, Po-Shen Loh, and Yufei Zhao. “The Critical Window for the Classical Ramsey-Turán Problem.” Combinatorica (October 22, 2014).
Version
Author's final manuscript
Abstract
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.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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DOI of Published Version
https://doi.org/10.1007/s00493-014-3025-3
