Maximizing fivecycles in $K_r$free graphs
Abstract
The Erdős Pentagon problem asks to find an $n$vertex trianglefree 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 $k \geq 3$, with the Turán graph begin the extremal graph for large enough $n$.
 Publication:

arXiv eprints
 Pub Date:
 July 2020
 arXiv:
 arXiv:2007.03064
 Bibcode:
 2020arXiv200703064L
 Keywords:

 Mathematics  Combinatorics
 EPrint:
 26 pages