Erdos in his Distinct distance Problem in a plane conjectured that the minimal number of distinct distance determined by $n$ points in a plane be $g(n)$,

$$g(n) \sim \frac{cn}{\sqrt{\log n}}$$

But for the special case which asks the minimum number of distinct distances that must be determined by $n$ points in a plane such that no $3$ are collinear be $g_{\textrm{no}3}(n)$, it is known that,

$$g_{\textrm{no}3}(n) \ge \left\lceil \frac{n-1}{3} \right\rceil$$

and $\displaystyle g_{\textrm{gen}}(n) \ge \left\lceil \frac{n-1}{3} \right\rceil$, where, $g_{\textrm{gen}}(n)$ asks the same question but for points in general position in a plane (i.e., no $3$ collinear and no $4$ conclyclic.)

Has there been any improvements on these particular casees of Erdos's Distance Problem?