I'm trying to understand a proof of a lemma in Erdos and Renyi's 1959 paper entitled "On random graphs I." I'll write what they've written first and then describe the quantities and my confusion. I'm stuck on a step in the proof in which the authors write:
"We obtain for $n \geq n_0,$ by using some elementary estimations, \begin{equation} \binom{n}{s}\frac{\binom{\binom{n}{2}-s(n-s)}{N_c}}{\binom{\binom{n}{2}}{N_c}} \leq \frac{e^{(3-2c)s}}{s!} \quad \textrm{for $s \leq \frac{n}{2}$ and} \end{equation} \begin{equation} \binom{n}{s}\frac{\binom{\binom{n}{2}-s(n-s)}{N_c}}{\binom{\binom{n}{2}}{N_c}} \leq \frac{e^{(3-2c)(n-s)}}{(n-s)!} \quad \textrm{for $s\geq \frac{n}{2}$ so, since} \end{equation} \begin{equation} P(\overline{E_M},n,N_c) \leq \sum_{M<s<n-\textstyle \frac{2N_c}{n}} \frac{\binom{n}{s} \binom{\binom{n}{2}-s(n-s)}{N_c}}{\binom{ \textstyle \binom{n}{2}}{N_c}} \textrm{,} \end{equation} \begin{equation} P(\overline{E_M},n,N_c) \leq \sum_{s>\frac{2N_c}{n}} \frac{e^{(3-2c)s}}{s!}+\sum_{s>M}\frac{e^{(3-2c)s}}{s!} \quad \textrm{for $n \geq n_0$."} \end{equation}
There $n$ is the number of vertices in any of the graphs being considered for the probability $P(\overline{E_M},n,N_c)$ that a graph randomly chosen from all of the graphs on $n$ vertices with $N_c=\textrm{int}(\frac{1}{2}n\log(n)+cn)$ edges is a member of the class $\overline{E_M}$ of graphs which satisfy the property that, for a "large number" (I don't understand what this actually means) $M$ the largest connected component in those graphs has fewer than $n-M$ vertices. $s$ is a quantity between $M$ and $n-\frac{2N_c}{2}$ introduced in the sum in the third equation above. Unless I'm just about blind (which is certainly possible,) $n_0$ is not introduced in the paper, and I have no clue what it means. My confusion is both about how the inequalities come about for the combinations (I've looked a little at Stirling's approximation and some identities on binomial coefficients and such things to no avail) and about how the bounds for the summations are transformed (why are $s \leq, \geq \frac{n}{2}$ chosen? What's $n_0$?)
If you've read this far and decided to help out and think that some more context might be necessary (or if you like the ideas and want to read the paper,) here is a PDF. I'd really appreciate any insights, thanks very much for reading.