Paper detail

Pentagons in triangle-free graphs

For all $n\ge 9$, we show that the only triangle-free graphs on $n$ vertices maximizing the number $5$-cycles are balanced blow-ups of a 5-cycle. This completely resolves a conjecture by Erdős, and extends results by Grzesik and Hatami, Hladký, Král', Norin and Razborov, where they independently showed this same result for large $n$ and for all $n$ divisible by $5$.