I was curious if there was a reference which answers the question, What is the maximum number of edges in a graph $G$ with $n$ vertices which does not contain a $5$-cycle? $k$-cycle? The analogous question for $k=4$ is well known.



Reference Request: Graph Edge Density
For cycles of odd length, the only extremal graphs for large $n$ are complete bipartite graphs with the sides as equal as possible. For smaller $n$ there can be other extremal graphs. The complete story was worked out fairly recently by [Füredi and Gunderson](http://arxiv.org/abs/1310.6766).


Cycles of even length are much harder. Some references are given in [this paper](https://www.researchgate.net/profile/Chunhui_Lai/publication/237145341_Some_open_problems_on_cycles_Journal_of_Combinatorial_Mathematics_and_Combinatorial_Computing_91_(2014)_51-64/links/0deec51b91cc6a88b3000000.pdf) and more in [this paper](http://arxiv.org/pdf/1306.5167.pdf). See [this paper](http://homepages.warwick.ac.uk/staff/O.Pikhurko/Papers/EvenCycle.pdf) too. Basically there are lots of partial results but the exact general solution is out of reach.


