Interpretation of probability statements in Nina Zubrilina's paper 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?
 A: I just skimmed the paper. When she writes

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

she means this inequality holds "asymptotically almost surely", i.e., the probability that this inequality holds goes to 1 as $n\to \infty$ (note: this is for a fixed $p$). This matches what you thought, and it's clear from her use of "a.a.s" in Lemma 2.3 and in Section 3. I previously misread your interpretation to think you were suggesting the result was a bound on $P(edim(G(n,p)))$ which of course makes no sense.
Similarly, in Lemma 2.2, edim$(G)$ refers to the family of graphs, and it's enough to prove the inequality holds a.a.s. as $n\to \infty$. So, when she says "Let $G = G(n,p)$ be the random graph" she is saying $G$ represents the family of graphs drawn from this random variable. That's confirmed in the proof, where she talks about the expected size of the edge set, $E$ -- it's a random variable. I think the definition of $\omega$ is fine as stated. She's choosing a number to guarantee a probabilistic inequality. It would be like if I said, "Consider an $n$-sided die, and let $n$ be a number such that the probability of rolling two consecutive 1s is less than 0.01." That statement does define a number. Of course, there is a probability on a given draw of $G = G(n,p)$ that you get a graph with no edges, or only 1 edge, and maybe she could have spelled out how to handle those cases. But I think they are vacuous, because the definition of "edge metric dimension" says "for any distinct $e_1,e_2\in E$". Also, the probability of this situation occurring goes to zero as $n\to \infty$.
