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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.

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    @Blatter, in this question $n \geq k+l$.2012-04-09

0 Answers 0