Israel Journal of Mathematics

, Volume 230, Issue 2, pp 613–652 | Cite as

The Ramsey–Turán problem for cliques

  • Clara Marie Lüders
  • Christian ReiherEmail author


An important question in extremal graph theory raised by Vera T. Sós asks to determine for a given integer t ≥ 3 and a given positive real number δ the asymptotically supremal edge density ft(δ) that an n-vertex graph can have provided it contains neither a complete graph Kt nor an independent set of size δn.

Building upon recent work of Fox, Loh and Zhao [The critical window for the classical Ramsey–Turán problem, Combinatorica 35 (2015), 435–476], we prove that if δ is sufficiently small (in a sense depending on t), then
$${f_t}(\delta ) = \{ \begin{array}{*{20}{c}} {\frac{{3t - 10}}{{3t - 4}} + \delta - {\delta ^2}iftiseven,} \\ {\frac{{t - 3}}{{t - 1}} + \delta iftisodd.} \end{array}$$


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© The Hebrew University of Jerusalem 2019

Authors and Affiliations

  1. 1.Fachbereich MathematikUniversität HamburgHamburgGermany

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