Maximizing five-cycles in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="d1e72" altimg="si9.svg"><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:msub></mml:math>-free graphs

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