Is this inequality true for Ramsey Numbers $R(n,n) - n \geq R(n,n-1)$

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Is this inequality true for Ramsey Numbers $R(n,n) - n \geq R(n,n-1)$. It seems like this is plausible, but i'm not really sure how to show it is or is not true. Ideas?

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It is not true for $n=2$ since $R(2,1)=1$ and $R(2,2)=2$. However, the statement holds for larger $n$. Have your tried anything ?