A *$k$-core* of a graph is a connected subgraph in which every vertex has degree at least $k$. Your set $S_k$ is the union of the vertex sets of the $k$-cores of the graph. 

https://en.wikipedia.org/wiki/Degeneracy_(graph_theory)#k-Cores

Similar to what Gerhard suggested, you can find the set $S_k$ by starting with $V(G)$ and deleting every vertex of degree less than $k$ until there are none left. 

With regards to $k$-cores in random graphs, one natural place to start might be  the paper [*Size and connectivity of the $k$-core of a random graph*][1] by Tomasz Łuczak.


  [1]: http://www.sciencedirect.com/science/article/pii/0012365X9190162U