Local maximum of inducibility profiles

For a graph $G$ and $e\in [0,1]$, denote by $I_G(e)$ the supremum of induced density of $G$ over $n$-vertex graphs with edge density $e$ as $n$ goes to infinity. Liu, Mubayi and Reiher asked if there exists a graph $G$, where $I_G(e)$ has a non-trivial local maximum. In this paper, we answer this problem affirmatively. We first show that $I_{K_{2,2,1}}(e)$ has at least two local maxima in $(0,1)$. Part of this proof is using flag algebras. Additionally, we determine $I_{K_{2,2,1}}(e)$, when $e=(k-1)\mathrm{/}k$ for every integer $k\ge 3.$ We also prove that $I_{K_t^{-}}(e)$ has a non-global local maximum for every $t\in \{5,8,11,\dots ,$ $74\}$. The proof combines a symmetrization theorem of Schelp and Thomason with Reiher’s clique density theorem.