Previously I have shown that for any positive integers $k,l$, and any real number $p\in (0,1)$, ramsey number $R(l,k) \geq n- {n\choose k} p^{{k \choose 2}} - {n\choose l} (1-p)^{{l \choose 2}}$.
Now I want to show, by using the above that $$R(3,k) \geq c (\frac{k}{log k})^a$$ for $a$ and $c$ positive constant.
I tried using stirling approximation i.e: from the first inequality above: $$ R(3,k) \geq n - (\frac{ne}{k})^k p^{k/2} - (\frac{ne}{3})^3 (1-p)^3$$
Now I thought of trying to do some analysis on the function: $$f(x)=1-Ax^{k-1}-Bx^2$$
But I believe this makes it more obsecure, and I don't see how to get out of this.
Any hints, answers, wishful thoughts? :-)
Thanks in advance.