For a finite abelian group $G$ and a subset $S\subseteq G$ with $0\in S=-S$, let 
 $$ \alpha(S):=\max\{|A|\colon(A-A)\cap S=\varnothing\} $$
and 
 $$ \omega(S):=\max\{|A|\colon A-A\subseteq S\}. $$ 
These quantities are the independence number and the clique number, respectively, of the Cayley graph induced by $S$ on $G$. Since Cayley graphs are vertex-transitive, [we have](https://books.google.co.il/books?id=P0XjCgAAQBAJ&pg=PA26&lpg=PA26&dq=clique-coclique+bound&source=bl&ots=oM3AV8H7qW&sig=O1Qb24ogPTDR6XvBlfJK7t_cVVs&hl=en&sa=X&ved=0ahUKEwjNmLKTvvXWAhVDIcAKHfY8B8oQ6AEIRzAF#v=onepage&q=clique-coclique%20bound&f=false) 
 $$ \alpha(S)\omega(S)\le |G|. \tag{$\*$} $$


Let now
 $$ \alpha^+(S):=\max\{|A|\colon(A+A)\cap S=\varnothing\} $$
and 
 $$ \omega^+(S):=\max\{|A|\colon A+A\subseteq S\}. $$ 
Is there an analog of the estimate ($\*$) for these quantities (maybe, a somewhat weaker one, or with some "restrictions apply" etc.)?



The product of the clique and independence numbers in the Cayley sum graph
If the set $S$ is arbitrary, then no strong variant of $(\*)$ holds unless $G$ has order $2$.


Indeed, consider a factorization of $G$, and suppose there is a factor $H \sim \mathbb{Z}\_n \neq \mathbb{Z}\_2$. Let $S$ be the set of all elements of $G$ such that their projection on $H$ is at least $n / 2$. But then $\alpha^+(S) \geq \frac{n - 1}{4n}|G|$ for $A = [0, \lfloor (n - 1) / 4 \rfloor]$. Also, $\omega^+(S) \geq \frac{\max(1, n / 4 - 3 / 2)}{n}|G|$ for $A = [\lceil n / 4 \rceil, \lfloor (n - 1) / 2 \rfloor]$. We can see that both $\alpha^+(S), \omega^+(S) = \Omega(|G|)$, hence LHS of $(\*)$ can be $\Omega(|G|^2)$, which is only a constant away from the trivial $|G|^2$ bound.


If $G \sim \mathbb{Z}\_2^k$, then $G$ has order 2, and $(\*)$ does hold since addition and subtraction are the same thing.


