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The key thing is to be precise about what model is used to generate your random graphs. It's typically not enough to pick out some set of events and specify that they hold with particular probabiliti …

Every graph of average degree $2k$ contains a subgraph of minimum degree $k$, so the threshold for the appearance of a non-empty $k$-core is at most $2k/n$. Since the threshold for the appearance of …

For fixed $k$, fixed $p$ and large $n$ I would expect this to be just the minimum possible diameter, scaling like $n^{1/k}$. In a random graph, any two sets of $\epsilon n$ vertices have the same numb …

My copy of the book being in my locked-down office makes it easy to avoid checking which section this question is from for a hint of the expected method, so here's a sledge hammer. Fix a partition $V …

Sudakov and Vu proved that, for $p \gg \log n / n$, $G_{n,p}$ has a perfect matching with high probability even after adversarially deleting $(1 - o(1))pn/2$ edges at each vertex. The same argument w …