Suppose $G$ is a group, $S \subset G$. Let’s call $S$ a Sidon subset iff $\forall$ quadruples $(a, b, c, d)$ of distinct elements of $S$ we have $ab \neq cd$ (named after Simon Sidon who studied such subsets in $C_\infty$). Let’s define $Sid(G)$ as the maximal possible cardinality of a Sidon subset in $G$. Do there exist such $0<c<C<+\infty$, such that $c \leq \frac{Sid(G)}{\sqrt{|G|}} \leq C$ for any finite group $G$?
The existence of $C$ can be proved the following way:
If $S$ is a Sidon subset of $G$, then $|G| \geq |S^2| \geq \frac{|S|(|S| - 1)}{2}$, thus $C = \sqrt{2} + 1$ will work.
The only thing I know about $c$ is that Erdos proved such constant exist for finite cyclic groups $G$. But does it exist for all finite $G$?
