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 f t (δ) that an n-vertex graph can have provided it contains neither a complete graph K t 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 ft(δ)={3t−103t−4+δ−δ2iftiseven,t−3t−1+δiftisodd.