From Turan's theorem we know that we can have $O(n^2)$ edges in a triangle-free graph. The theorem talks about $k$-free graphs, where the graph doesn't contain any cliques of size $k$.
I am looking for an analogy for $k$-cycle-free graphs, that is a graph that doesn't contain a cycle of size $k$ or smaller. All cycles must be of size at least $k+1$.
For $k = 3$, the two definitions are the same (a triangle is a clique and a cycle). So even help with $k=4,5$ will help me a lot.