Maximizing five-cycles in $K_r$-free graphs

Abstract

The Erdős Pentagon problem asks to find an $n$-vertex triangle-free graph that is maximizing the number of $5$-cycles. The problem was solved using flag algebras by Grzesik and independently by Hatami, Hladký, Král', Norin, and Razborov. Recently, Palmer suggested the general problem of maximizing the number of $5$-cycles in $K_{k+1}$-free graphs. Using flag algebras, we show that every $K_{k+1}$-free graph of order $n$ contains at most \[\frac{1}{10k^4}(k^4 - 5k^3 + 10k^2 - 10k + 4)n^5 + o(n^5)\] copies of $C_5$ for any $k\ge 3$, with the Turán graph begin the extremal graph for large enough $n$.

26 pages