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More precisely, we prove that for sufficiently large $n$, $$ \\alpha(G(\\mathcal K_n)) \\geq f_0(\\mathcal K_n) - \\frac{C\\,f_0(\\mathcal K_n)}{\\left(\\log f_0(\\mathcal K_n)\\right)^{\\lfloor d/2 \\rfloor-1}},$$ where $C = C(d) > 0$. 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