For finite graphs there exists a graph $G$ on $n$ vertices with $\delta(G)\geqslant \lfloor n^2/4\rfloor$. Namely, let $G$ be the disjoint union of two cliques $C_1,C_2$ on $a=\lceil n/2\rceil$ and $b=\lfloor n/2\rfloor$ vertices resp. The vertices of $C_1$ should correspond to disjoint sets $X_1,\ldots,X_a$, the vertices of $C_2$ to disjoint sets $Y_1,\ldots,Y_b$. Any two sets $X_i,Y_j$ should have a common element $p_{ab}$, and they are all distinct since $X_i$'s are disjoint aswell as $Y_j$'s. Therefore the ground set $S$ must contain at least $ab=\lfloor n^2/4\rfloor$ elements.
If I remember well, any edge set of the graph on $n$ vertices is a disjoint union of at most $\lfloor n^2/4\rfloor$ cliques (cliques of size 2 and 3 should be enough), that implies $\delta(G)\leqslant \lfloor n^2/4\rfloor$. So this estimate for fixed $n$ is sharp.