The $k$-core of a finite graph is defined as follows. Delete all vertices of degree $< k$ and repeat until there are no such vertices left. If there is a nonempty subgraph remaining, necessarily of minimum degree $\ge k$, we call this graph the $k$-core. If the deletion process results in removing all vertices, we say that the graph has no $k$-core.

Let $f(n,k)$ denote the maximum number of edges in a simple graph on $n$ vertices with no $k$-core. For example, if $f(n,2)=n-1$ for every $n$, since $n$ edges are sufficient (and necessary) to guarantee the existence of a cycle.

What about $k \ge 3$? Maybe a formula is too much to hope for, so for fixed $k$ what are the asymptotics of $f(n,k)$ as $n \to \infty$?

I suspect this kind of question has been studied, so any references would be appreciated. It is very similar in flavor to questions in "classical" extremal graph theory, where one tries to maximize the number of edges in a graph on $n$ vertices with no $H$-subgraph, where $H$ is a fixed graph. This question fits in a more general setting where $H$ ranges over an infinite family of possible subgraphs.