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Let $r(S(n,m))$ denote the Ramsey number of the double star $S(n,m)$.\n  In 1979 Grossman, Harary and Klawe have shown that $$r(S(n,m)) = \\max\\{n+2m+2,2n+2\\}$$ for $3 \\leq m \\leq n\\leq \\sqrt{2}m$ and $3m \\leq n$. They conjectured that equality holds for all $m,n \\geq 3$. Using a flag algebra computation, we extend their result showing that $r(S(n,m))\\leq n+ 2m + 2$ for $m \\leq n \\leq 1.699m$. 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