Cashman–Kelley’s odd-cycle Cayley–Turán conjecture

Let $p$ be an odd prime, let $\mathbb Z_p$ denote the residue classes modulo $p$, and let $S\subseteq \mathbb Z_p$ be symmetric, with $S=-S$ and $0\notin S$. Write $\operatorname{Cay}(\mathbb Z_p,S)$ for the undirected Cayley graph whose vertices are $\mathbb Z_p$, with $x,y$ adjacent exactly when $x-y\in S$. For a graph $F$, define

$$\operatorname{ex}_{\operatorname{Cay}}(F,\mathbb Z_p)=\max\{|S|:S=-S,\ 0\notin S,\ \operatorname{Cay}(\mathbb Z_p,S)\text{ contains no copy of }F\}.$$

Cashman–Kelley’s conjecture. For $p>2\ell+1$, the Cayley–Turán number of the odd cycle $C_{2\ell+1}$ satisfies

$$\operatorname{ex}_{\operatorname{Cay}}(C_{2\ell+1},\mathbb Z_p)=2\left\lfloor\frac{p+2\ell+1}{2(2\ell+1)}\right\rfloor.$$

Cashman and Kelley initiated the study of Cayley–Turán numbers and proved the corresponding formula $\operatorname{ex}_{\operatorname{Cay}}(C_{2\ell},\mathbb Z_p)=2$ for even cycles when $2\leq\ell<p/2$; the displayed odd-cycle formula remains the conjectured case.