The Maximum Number of Three Term Arithmetic Progressions, and Triangles in Cayley Graphs

Let $G$ be a finite Abelian group. For a subset $S \subseteq G$, let $T_3(S)$ denote the number of length three arithemtic progressions in $S$ and Prob[$S$] $= \frac{1}{|S|^2}\sum_{x,y \in S} 1_S(x+y)$. For any $q \ge 1$ and $\alpha \in [0,1]$, and any $S \subseteq G$ with $|S| = \frac{|G|}{q+\alpha}$, we show $\frac{T_3(S)}{|S|^2}$ and Prob[$S$] are bounded above by $\max\left(\frac{q^2-\alpha q+\alpha^2}{q^2},\frac{q^2+2\alpha q+4\alpha^2-6\alpha+3}{(q+1)^2},\gamma_0\right)$, where $\gamma_0 < 1$ is an absolute constant. As a consequence, we verify a graph theoretic conjecture of Gan, Loh, and Sudakov for Cayley graphs.