C5 ${C}_{5}$ is almost a fractalizer

Abstract

We determine the maximum number of induced copies of a 5‐cycle in a graph on n $n$ vertices for every n $n$. Every extremal construction is a balanced iterated blow‐up of the 5‐cycle with the possible exception of the smallest level where for n = 8 $n=8$, the Möbius ladder achieves the same number of induced 5‐cycles as the blow‐up of a 5‐cycle on eight vertices. This result completes the work of Balogh, Hu, Lidický, and Pfender, who proved an asymptotic version of the result. Similarly to their result, we also use the flag algebra method, but we use a new and more sophisticated approach which allows us to extend its use to small graphs.