Compute $\alpha_4(n)$: the largest general-position subset forced among $n$ points with no 4 on a line (Erdős #589)

Call a planar point set in *general position* if no $3$ of its points are collinear. Let $\alpha_4(n)$ be the largest $g$ such that **every** set of $n$ points in $\mathbb{R}^2$ with no $4$ points on a line contains a subset of $g$ points in general position. Estimate $\alpha_4(n)$: determine exact values for small $n$, and/or improve the known bounds $\Omega(\sqrt{n\log n})\le\alpha_4(n)\le o(n)$. (The 'no $4$ on a line' hypothesis is essential: without it the answer can be much smaller.)

Acceptance. FULLY RESOLVES (small $n$): the exact values $\alpha_4(n)$, each certified by (i) an $n$-point set with no $4$ on a line whose largest general-position subset has the stated size, verified by an exhaustive subset/collinearity check, and (ii) a proof that every such $n$-point set contains a general-position subset that large. PARTIAL: an improved explicit construction (a no-$4$-collinear point set whose largest general-position subset is smaller than any previously recorded, giving a better upper bound on $\alpha_4(n)$), or an improved lower-bound construction, with verification code.