On the Number of Pentagons in Triangle-Free Graphs
  Using the formalism of flag algebras, we prove that every triangle-free graph $G$ with $n$ vertices contains at most $(n/5)^5$ cycles of length five. Moreover, the equality is attained only when $n$ is divisible by five and $G$ is the balanced blow-up of the pentagon. We also compute the maximal number of pentagons and characterize extremal graphs in the non-divisible case provided $n$ is sufficiently large. This settles a conjecture made by Erd\H{o}s in 1984. 
Hamed Hatami, Jan Hladk\'y, Daniel Kr\'al, Serguei Norine, Alexander   Razborov
math.CO