I am trying to understand the upper bound that David Conlon produced for the diagonal Ramsey Numbers
$$R(n+1,n+1) \leq n^{-c \frac { log n}{log \;{log n}}} \binom {2n}{n}$$
With the binomial theorem it is shown that
$$R(n+1,n+1) \leq \frac{4^n} {\sqrt {\pi n}}$$
which is not difficult to understand. I'm having a hard time trying to understand the term
$$n^{-c \frac { log n}{log \;{log n}}}$$
and how big of an improvement it is.