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This gives an alternative proof of Shearer's upper bound on the Ramsey number $R(3,k)$. We then prove that the total number of independent sets in a triangle-free graph with maximum degree $d$ is at least $\\exp \\left[\\left(\\frac{1}{2}+o_d(1) \\right) \\frac{\\log^2 d}{d}n \\right]$. 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This gives an alternative proof of Shearer's upper bound on the Ramsey number $R(3,k)$. We then prove that the total number of independent sets in a triangle-free graph with maximum degree $d$ is at least $\\exp \\left[\\left(\\frac{1}{2}+o_d(1) \\right) \\frac{\\log^2 d}{d}n \\right]$. 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