The Sylvester--Gallai dimension of graphs

For an undirected graph $G$, its \emph{Sylvester–Gallai dimension} $\text{SGdim}(G)$ is the largest affine dimension of a configuration of distinct real points indexed by $V(G)$ in which every line determined by an edge of $G$ is a special line (contains at least three points). Hence, the classical Sylvester–Gallai theorem can be stated as $\text{SGdim}(K_n)=1$ for the complete graph $K_n$. We initiate the systematic study of this new graph parameter and prove lower bounds for certain graph families (bounded degree, sparse, minor-free) as well as an upper bound for random graphs.