A Turán Theorem for Cayley Graphs

In this note, we give a Tur\'an theorem for Cayley graphs $\Cay(\Z_p,S)$ over prime cyclic groups $\Z_p$. For a graph $F$ and a finite abelian group $G$, define the Cayley--Tur\'an number by $$\exCay(F,G) = \max\{|S|:S=-S\subseteq G\setminus\{0\},\ \Cay(G,S)\text{ is }F\text{-free}\}.$$ Using a polynomial method, we prove that for every odd prime $p$ and every $1\le r\le p-1$, $$\exCay(K_{r+1},\Z_p) = p-1-2\left\lfloor\frac{p}{r+1}\right\rfloor .$$ The extremal construction is the complement of the short-difference interval $$D_0=\{0,\pm1,\ldots,\pm\lfloor p/(r+1)\rfloor\}.$$ 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.