The Pentagon Problem of Erdős 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 a more 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 $\frac{1}{10k^4}(k^4-5k^3+10k^2-10k+4)n^5+o\left(n^5\right)$ copies of $C_5$ for any $k\ge 3$, with the Turán graph being the extremal graph for large enough $n$.

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最大化五个周期 ${钾}_r$-免费图表

Erdős 问题的五角大楼问题要求找到一个 $n$- 无顶点三角形图，使 5 个循环的数量最大化。该问题由 Grzesik 使用标志代数解决，由 Hatami、Hladký、Král'、Norin 和 Razborov 独立解决。最近，Palmer 提出了一个更一般的问题，即最大化 5 个循环的数量。${钾}_{克+1}$-免费图表。使用标志代数，我们证明每个${钾}_{克+1}$- 订单的自由图 $n$ 最多包含 $\frac{1}{10{克}^4}({克}^4-5{克}^3+10{克}^2-10克+4)n^5+○\left(n^5\right)$ 副本 $C_5$ 对于任何 $克\ge 3$，图兰图是足够大的极值图 $n$.