Hello!

**Background:** I am trying to understand a proof in

Béla Bollobás, Random Graphs, 2nd edition, Ch. 11.1

about the clique number of G(n,p) random graphs.

**The problem:** At some point, the following sequence is defined:

$F_\ell := \frac{\binom{r}{\ell} \binom{n-r}{r - \ell} } {\binom{n}{r}} (p^{-\binom{\ell}{2}} - 1)$

Afterwards, it is said that

"Routine and crude calculations show that, if $n$ is sufficiently large and $3 \leq \ell \leq r-1$, then $F_\ell < F_3 + F_{r-1}$"

It looks like $F_\ell$ is first decreasing and then increasing, such that it is either bounded by $F_3$ or $F_{r-1}$. However, I do no not see how to show this *formally*? Especially in the context, where $r = r(n) = \mathcal{O}(n^{\frac13})$ and $p = p(n), 0 < p < 1$ such that

$\binom{n}{r}p^{\binom{r}{2}} \to \infty$ and $\binom{n}{r+1}p^{\binom{r+1}{2}} \to 0$ for $n \to \infty$ holds (as stated in Theorem 11.6 by Bollobas).

In fact, it seems very difficult to say something about these expression as $p(n)$ is specified only implicit by these expectations and $r$ is also depending on $n$.

Does anyone have an idea?