$ \omega_i= w_i/n$ usually is the fraction of variables belonging to $W_i.$ And $\lambda_i\,= l_i\,/ (km)=l_i/(k\gamma n)$ is the fraction of slots filled with a variable from $W_i.$ We use $\bar{\omega} = (\omega_0, \omega_1, \omega_2), $ and $\bar{\lambda}=(\lambda_0, \lambda_1, \lambda_2).$ Sometimes $\omega_i, \lambda_i$ stand for arbitrary reals, this should be clear form the context.

 Outline of the proof of Theorem \[EX2EI\] 
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First, bounds for $M(m, n)$ and $K(\bar{l}).$ We consider $q(x):= \exp(x)-x-1=  \sum_{j \ge 2} \frac{x^j}{j!} $ for $x\ge 0.$ Then for $a>0$ and all $m, n$ $$\begin{aligned}
M(m, n) \,= \, \mbox{Coeff}[x^m, q(x)^n]\cdot m! \, < \, q(a)^n \cdot \frac{1}{a^m } \cdot m!\,\le \,  q(a)^n \left(\frac{m}{a\cdot e}\right)^m \cdot O(\sqrt{m})  \label{M1}\end{aligned}$$ using Stirling in the form $ m! < (m/e)^m \cdot  O(\sqrt{m}).$

To get rid of the $\sqrt{m}-$factor we let $Q(x):= xq'(x)/q(x)$ with $q'(x)$ the derivative of $q(x), \,q'(x)=\exp(x)-1$ for $x>0.$ Then $Q'(x)>0$ for $x>0,$ $Q(x)>x,$ and $Q(x)\longrightarrow 2$ for $x \longrightarrow 0.$ Thus, for $y>2$ the inverse function $Q^{-1}(y)>0$ is defined and differentiable. Lemma \[MLOC\] is proved in Section \[REPRO\].

\[MLOC\] Let $Cn \ge m \ge (2+\varepsilon)n, \, \, C, \varepsilon>0$ constants. Then $$\begin{aligned}
M(m, n) \,= \, \Theta(1)\cdot \left( \frac{ m}{ae}\right)^m\cdot q(a)^n \mbox{ with } a \mbox{ 