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Claim. Every set of 10 points in the plane in general position contains five points in convex position with no point of the set inside their convex hull, and there is a set of 9 points in general position with no such pentagon. In the notation of Problem 216, $g(5)=10$.

Covers. The value $g(5)=10$, the instance $k=5$ of the estimation half of the question. The value $g(4)=5$ is Erdős's observation; the instance $k=6$ is settled on the Nicolás, Gerken and Heule–Scheucher claim pages, and $g(k)$ does not exist for $k\ge7$ by Horton's result.

Acceptance. The paper is refereed: H. Harborth, Konvexe Fünfecke in ebenen Punktmengen, Elemente der Mathematik 33 (1978), 116–118, in the journal's Kleine Mitteilungen; the link above is the e-periodica record of the volume. The site's remarks credit the value to this paper; the site's label, disproved, rests on Horton's result, so that remark is not acceptance of this partial claim.