Speaker: Kyle Murphy
Title: Maximizing five-cycles in $K_{k+1}$-free graphs.
Abstract:
Using flag algebras, we show that every $K_{k+1}$-free graph of order $n$ contains at most
$$\frac{1}{k^4}(12k^4 - 60k^3 + 120k^2 - 120k + 48)\binom{n}{5}$$
copies of $C_5$ for any $k \geq 3$, with the Tur\'{a}n graph giving an asymptotically tight lower bound. We also provide a stability result. This problem was suggested by Cory Palmer, and extends the work of Grzesik and independently Hatami Hladk\'{y}, Kr\'{a}l', Norin, and Razborov, who proved a similar result for triangle free graphs. Additionally, we apply a technique developed by Lidick\'{y} and Pfender to show that for $k = 4,5 $ and $6$, the Tur\'{a}n graph is the unique extremal graph for all $n$. In this talk, I will also give a brief introduction to the flag algebra method, which was developed by Razborov. This talk is based on joint work with Bernard Lidick\'{y}.