Using Razborov's flag algebras we show that a triangle-free graph on n vertices contains at most (n/5)^5 cycles of length five. It settles in the affirmative a conjecture of Erdos.
@article{arxiv.1102.0962,
title = {On the maximum number of five-cycles in a triangle-free graph},
author = {Andrzej Grzesik},
journal= {arXiv preprint arXiv:1102.0962},
year = {2012}
}