The optimality conjecture for induced odd cycles in planar graphs

Let $k\geq 3$, and let $h_k(n)$ denote the number of induced copies of $C_{2k+1}$ obtained by evenly blowing up $k$ pairwise non-adjacent vertices in a $C_{2k+1}$ on $n$ vertices. For a planar graph $G$ on $n$ vertices, let $\mathscr{H}(G,C_{2k+1})$ be the number of induced copies of $C_{2k+1}$ in $G$. Optimality conjecture for induced odd cycles. The maximum of $\mathscr{H}(G,C_{2k+1})$ over all planar graphs $G$ on $n$ vertices is $h_k(n)$. This construction gives the best currently known lower bound for the inducibility of odd cycles in planar graphs; its optimality remains open for all $k\geq 3$.