I asked this question on Math.stackexchange but got no answer.

In the paper Zubrilina - Asymptotic behavior of the edge metric dimension of the random graph (MR, the main result is

$$\operatorname{edim}(G(n,p)) \leq (1+o(1))\frac{4 \log n}{\log(1/q)},$$ where $q= 1-2p(1-p)^2(2-p)$.

My **first question** is how should I interpret the result, what is $\operatorname{edim}$ of random graph. Should I interpret it as
$$\mathbb{P}\left[\operatorname{edim}(G(n,p)) \leq (1+o(1))\frac{4 \log n}{\log(1/q)}\right] \rightarrow 1 \text{ as $n \rightarrow \infty$?} $$

My second question is concerned with how to interpret lemma 2.2, it is stated that

Let $G=G(n,p)$ be the random graph. Let $V,E$ denote the vertex and edge sets. Let $\omega \in \{1,\dotsc,n\}$ be such that for any two distinct edges $e_1$, $e_2$ of $E$, a uniformly random subset $W \subset V$ of size $\omega$ satisfies $$\mathbb{P}( \text{$W$ does not distinguish $e_1$, $e_2$}) \leq 1/n^4p^2. $$ Then $$\operatorname{edim}(G) \leq \omega.$$

So, firstly how should I understand $E$ as subset of a random graph, and how can I fix two edges of this seemingly random set by saying "for any two distinct edges $e_1,e_2 \in E$". I am confused about how I interpret such statement. Can any one clarify them?