A Turán Theorem for Cayley Graphs

In this note, we give a Turán theorem for Cayley graphs $\text{Cay}({\mathbb{Z}}_p,S)$ over prime cyclic groups ${\mathbb{Z}}_p$. For a graph $F$ and a finite abelian group $G$, define the Cayley–Turán number by $$\text{exCay}(F,G)=\max \{\mathrm{\mid }S\mathrm{\mid }:S=-S\subseteq G\setminus \{0\},\text{Cay}(G,S)\text{ is }F\text{-free}\}\mathrm{.}$$ Using a polynomial method, we prove that for every odd prime $p$ and every $1\le r\le p-1$, $$\text{exCay}(K_{r+1},{\mathbb{Z}}_p)=p-1-2\lfloor \frac{p}{r+1}\rfloor \mathrm{.}$$ The extremal construction is the complement of the short-difference interval $$D_0=\{0,\pm 1,\dots ,\pm \lfloor p\mathrm{/}(r+1)\rfloor \}\mathrm{.}$$ We also discuss what changes for general finite abelian groups, showing why the exact prime-cyclic formula does not extend verbatim to composite cyclic groups.