For an irrational $\alpha$, we can define a pre-dimension on the class of graphs given by:
$$\delta_\alpha(G)=|V(G)|-\alpha|E(G)|.$$
The idea of the Hrushovski ab-initio construction is to restrict to the class $\mathcal{C}_\alpha$ of those graphs $A$ such that for every $A'\subseteq A$, $\delta_\alpha(A')\geq 0$}, to then build an infinite model $\mathcal{M}_\alpha$ using some amalgamation construction.
It is known that when $\alpha\in (0,1)$ is irrational, the limit model $\mathcal{M}_\alpha$ is stable but not superstable, and this should boil down to the fact that there is an infinite sequence $A_1\subseteq A_2\subseteq \cdots$ of graphs in $\mathcal{C}_\alpha$ such that $\delta_\alpha(A_i)>\delta_\alpha(A_{i+1})$ for every $i\in\mathbb{N}$.
Question 1: How can we prove there is such a sequence of finite graphs?
Question 2: For a given $\alpha\in (0,1)$ irrational, is it possible to construct such a sequence explicitly?
So far we have used some equidistribution theorems to show that there must be pairs $(n_i,m_i)$ of integers (thought of as pairs (number of vertices, number of edges) such that
$$n_1-\alpha\cdot m_1 > n_2-\alpha\cdot m_2>\cdots>0,$$
but we have not been able to ensure that there are graphs with these given number of vertices and edges, that belong to the class $\mathcal{C}_\alpha$.
An explicit construction is given in the proof of Lemma 4.1 in Laskowski's paper A simpler axiomatization of the Shelah-Spencer almost sure theories (pdf). The forking chain witnessing non-superstability is constructed in Proposition 7.6, as a direct application of Lemma 4.1.