A subquadratic bound for generalized Turán numbers of odd cycles

Abstract: For a graph $H$ and a family of graphs $\mathcal F$, let $\text{ex}(n,H,\mathcal F)$ denote the maximum number of copies of $H$ in an $\mathcal F$-free graph on $n$ vertices. For every integer $i\ge 3$, let $C_i$ denote the cycle of length $i$. For $r\ge 3$, set $\mathscr {C}<em>r={C_3,C_4,\ldots,C_r},$ and set $\mathscr {C}_2=\varnothing$. In this paper, we prove that, for all integers $l&gt;k\ge 2$, $$ \text{ex}(n,C</em>{2k+1},\mathscr {C}<em>{2k}\cup{C</em>{2l+1}}) =O_{k,l} \left(n^{{2-\frac{1}{k(k+1)(l-k)}}\} \ \right). $$ Together with the known upper bounds for the number of triangles in $C_{2l+1}$-free graphs, this confirms a conjecture of Gerbner, Győri, Methuku, and Vizer.