I'm taking a course which follows the book:
High-Dimensional Probability by Roman Vershynin.
There is an exercise (2.4.4) in the book which I have trouble with:
Consider a random graph $G \sim G(n,p)$ with expected degrees $d=o(\log n)$. Show that with high probability (say 0.9), $G$ has a vertex with degree $10d$.
To solve this I want to take a subset of vertices $V' \subset V$ s.t. each two vertices in $V'$ are not neighbors in the graph, so their degrees are independent. Then use Poisson approximation.
But I think that this way the expected degree of the vertices in this group is no longer $d$, and I cant figure out what can I say about $|V'|$?